Assessment of the local stress state through macroscopic variables

101098rsta20031172

Assessment of the local stress statethrough macroscopic variables

By Robert P Lipton

Department of Mathematics Louisiana State UniversityBaton Rouge LA 70803 USA

Published online 27 March 2003

Macroscopic quantities beyond enotective elastic tensors are presented that can be usedto assess the local state of stress within a composite in the linear elastic regime Theseare presented in a general homogenization context It is shown that the gradient ofthe enotective elastic property can be used to develop a lower bound on the maximumpointwise equivalent stress in the shy ne-scale limit Upper bounds are more sensitiveand are correlated with the distribution of states of the equivalent stress in the shy ne-scale limit The upper bounds are given in terms of the macrostress modulationfunction This function gauges the magnitude of the actual stress For 1 6 p lt1 upper bounds are found on the limit superior of the sequence of Lp norms ofstresses associated with discrete microstructure in the shy ne-scale limit Conditionsare given for which upper bounds can be found on the limit superior of the sequenceof L1 norms of stresses associated with the discrete microstructure in the shy ne-scalelimit For microstructure with oscillation on a sumacr ciently small scale we are able togive pointwise bounds on the actual stress in terms of the macrostress modulationfunction

Keywords composite materials elastic stress analysis microstructure

1 Introduction

Several technologically important applications have beneshy ted from the use of gradedcomposite materials (see Glasser 1997 Koizumi 1997) Graded materials provide thedesigner with the freedom to tailor the materialrsquos microstructure in order to enhancestructural performance In addition to designing for structural response it is of vitalimportance to control the stress inside structural components made from compositematerials Regions of large stresses are most likely to be the shy rst to exhibit failureduring service

Failure criteria such as the yield surface in plasticity require the knowledge of thestresses within each phase of the composite However graded composite materialsare often made up of a very large number of discrete phases interspersed on a smallscale The direct evaluation of stress shy elds resulting from body or boundary loads liesbeyond current experimental methods A central theoretical problem understandingthe asymptotic behaviour of the stress for linearly elastic composites in the limit ofinshy nitely shy ne mixing of the discrete phases The goal of this analysis is to identify

One contribution of 12 to a Theme `Micromechanics of deguid suspensions and solid compositesrsquo

Phil Trans R Soc Lond A (2003) 361 921946

921

cdeg 2003 The Royal Society

922 R P Lipton

suitable macroscopic quantities that can be used to assess the state of stress withina composite

In this treatment no assumption on the microstructure is made and the homoge-nized formula for the mean-square stress reguctuation is obtained (see theorem 23) Itis shown that the formula provides a lower bound on the limit inferior of the sequenceof L 1 norms of stresses associated with the discrete microstructure in the shy ne-scalelimit (see theorem 24) Mean-square stress reguctuations appear in many contextsincluding bounds for nonlinear composites (Ponte Casta~neda amp Suquet 1998) andestimates for failure surfaces for statistically homogeneous random composites (Bury-achenko 2001)

To obtain upper bounds on Lp norms 1 6 p 6 1 new quantities for the assessmentof the actual stress are introduced see (15) (34)(37) and (49) These quantitiesare referred to as macrostress modulation functions They measure the amplishy ca-tion or diminution of the macrostress by the underlying microgeometry For simplehypotheses on the asymptotic distribution of states for the stress these quantitiesare used to obtain upper bounds on the limit superior of the sequence of L1 norms ofstresses associated with the discrete microstructure in the shy ne-scale limit (see corol-laries 35 and 46) For 1 6 p lt 1 corollaries 36 and 47 provide upper bounds onthe limit superior of the sequence of Lp norms of stresses associated with the discretemicrostructure in the shy ne-scale limit For sumacr ciently small pointwise bounds onthe actual stress are given in terms of the macrostress modulation function (see (115)and corollaries 34 and 45) These results follow from a homogenization constraintthat correlates the macrostress modulation functions with the homogenized distribu-tion of states for the stress (see theorems 31 and 42) We apply the homogenizationconstraint to assess the size of sets over which the local stress in the compositeexceeds a prescribed value (see corollaries 32 33 43 and 44) The principal toolused to derive the constraint is a local formula for the derivative of enotective prop-erties given in theorem 22 The methods developed here can be applied to obtainanalogous results for local regux and gradient shy elds in the contexts of heat conductionDC electric transport and problems of dinotusion

Composites made from N linearly elastic materials are considered The domaincontaining the composite is denoted by laquo and the stress tensor inside the compositeat the point x is denoted by frac14 (x) Failure criteria are often given in terms of a co-ordinate invariant measure of the stress In this treatment we consider the equivalentstress given by

brvbar ( frac14 ) = brvbar frac14 frac14 (11)

where brvbar is a positive deshy nite fourth rank tensor (see Tsai amp Hahn 1980) Examplesof (11) include the Von Mises equivalent stress given by

brvbar ( frac14 ) = 12[( frac14

11 iexcl frac14 22)2+(frac14


11 iexcl frac14 33)2]+3(( frac14

12)2+(frac14 13)2+(frac14

23)2) (12)

and the magnitude of the stress given by brvbar ( frac14 ) = j frac14 j2 =P3

ij = 1( frac14 ij)2 The set in

the ith phase where brvbar ( frac14 ) gt t is denoted by Sti The stress distribution function

para i (t) is the volume (measure) of S

ti One of the principal results of this paper istheorem 31 It provides a constraint relating macroscopic stress modulation functionand S

ti in the limit of vanishing To motivate theorem 31 and the macrostress modulation function we consider a

composite with uniform periodic microstructure contained in the unit cube Q The

Phil Trans R Soc Lond A (2003)

Assessment of the local stress state 923

local elastic tensor for a Q-periodic conshy guration of N anisotropic elastic materialsis denoted by C(y) Each phase has an elastic tensor specishy ed by Ai and C(y) = Ai

in the ith phase The characteristic function of the ith phase is written as Agrave i(y) andtakes a value of unity in the ith phase and zero otherwise No constraints are placedon the arrangement of the phases inside Q Here the description of the enotective elasticproperties and local stress and strain are well known (see Milton 2002) For this casethe strain deg (y) in the composite can be decomposed into the average strain middotdeg and aperiodic reguctuation deg (w(y)) where the periodic displacement w is the solution of

iexcl div(C(y)( deg (w(y)) + middotdeg )) = 0 (13)

The stress in the composite is frac14 (y) = C(y)( deg (w(y)) + middotdeg ) The average stress isrelated to the average strain middotdeg by middotfrac14 = CEmiddotdeg Here CE is the enotective elastic tensor

One is interested in the maximum equivalent stress in the ith phase generated bysubjecting Q to the average stress middotfrac14 Or more generally it is of interest to knowwhether the specimen is experiencing an equivalent stress greater than a prescribedvalue on a subdomain of non-zero volume Thus a natural choice for stress assessmentis the L1 (Q) norm The L 1 (Q) norm of a function g denoted by kgkL 1 (Q) is deshy nedin the following way Consider the set of points St in Q where jg(y)j gt t The L 1 (Q)norm of g is deshy ned to be the greatest lower bound on the parameter t for which St

has zero volume (measure) The volume (measure) of the set St is denoted by jStjIt is evident that if t gt kgkL 1 (Q) then jStj = 0 and if kgkL 1 (Q) gt t then jStj gt 0The L 1 norm is consistent with failure criteria that require the equivalent stress toexceed a critical value over a shy nite volume of the composite sample before it will fail

The tensor F i(y) is deshy ned for every 3 pound 3 tensor middotdeg by

F i(y)middotdeg middotdeg = Agrave i(y) brvbar ( frac14 (y)) = Agrave i(y)brvbar(Ai( deg (w(y)) + middotdeg )) (Ai( deg (w(y)) + middotdeg )) (14)

Setf i(SEmiddotfrac14 ) = kF i(y)SEmiddotfrac14 SEmiddotfrac14 kL(Q)1 = k Agrave i(y) brvbar ( frac14 (y))kL(Q) 1 (15)

where SE is the enotective compliance (CE)iexcl1 The quantity f i(SEmiddotfrac14 ) is referred to asthe macrostress modulation function and measures the amplishy cation or diminutionof middotfrac14 by the microstructure Next we consider k = 1=k periodic composites specishy edby the re-scaled elastic tensors fC(x=k)g 1

k = 1 The Q-periodic solution wk of

iexcl div(C(x=k)( deg (wk (x)) + middotdeg )) = 0 (16)

is given by wk (x) = kw(x=k) and frac14 k (x) = frac14 (x=k) For every k-periodic com-posite it follows from the change of variables y = x=k that the modulation of theaverage stress by the microstructure is always given by f i(SEmiddotfrac14 ) ie

f i(SEmiddotfrac14 ) = k Agrave i(x=k) brvbar ( frac14 k (x))kL(Q)1 (17)

and the stress distribution function is independent of k ie

para k

i (t) = para 1i (t) = sup3 ti (18)

It is evident that for this special case the modulation of the average stress by themicrostructure reduces to an analysis of the unit periodic composite For this casethe homogenization constraint expressed in theorem 31 follows immediately fromthe deshy nition of f i(SEmiddotfrac14 ) Indeed from (15) it is clear that t gt f i(SEmiddotfrac14 ) implies


924 R P Lipton

that sup3 ti = 0 and equivalently if sup3 ti gt 0 it follows that f i(SEmiddotfrac14 ) gt t This deliversthe homogenization constraint given by

sup3 ti(fi(SEmiddotfrac14 ) iexcl t) gt 0 (19)

This is the specialization of theorem 31 to the case at hand In the general contextthe macroscopic stress is not uniform and the microgeometry is not periodic and canchange across the specimen However theorem 31 shows that a constraint of theform (19) holds and is given in terms of quantities ~f i and sup3 ti(x) that generalizef i(SEmiddotfrac14 ) and sup3 ti

To shy x ideas and to give further motivation for theorem 31 we provide a generaliza-tion of (19) that holds for a composite with uniform periodic microstructure that issubjected to a non-uniform load Here the uniformly periodic composite specimen laquois subjected to a body load f The displacement uk vanishes on the specimen bound-ary The k-periodic microstructure is given by the re-scaling Ck (x) = C(x=k) Thestress and displacement in the specimen are denoted by frac14 k and uk respectivelyThe equation of elastic equilibrium is given by

iexcl div frac14 k = f (110)

The elastic strain deg (uk ) is related to the stress by frac14 k = Ck deg (uk )The fundamental feature that distinguishes this example from the previous one

is that the sets Sk

ti are no longer periodic and self similar The self similarity isbroken by the fact that the non-uniform load f and the specimen shape can beincommensurate with the periodicity of the microgeometry

For the case at hand the constraint analogous to (19) is written in terms of themacroscopic or homogenized stress From the theory of periodic homogenization(Bensoussan et al 1978) one has that frac14 k and deg (uk ) converge to the macroscopicstress frac14 M and strain deg (uM ) as k goes to zero The macroscopic displacement u M van-ishes on the boundary and the macroscopic stress satisshy es the equilibrium equationiexcl div frac14 M = f The stress and strain are related through the homogenized constitu-tive law

frac14 M (x) = CE deg (u M (x)) (111)

Now substitute frac14 M (x) for middotfrac14 in (15) to write f i(SE frac14 M (x)) In order to complete thedescription of the homogenization constraint a suitable generalization of sup3 ti givenby (18) is needed The indicator function for the set Sk

ti is written Agrave k

ti taking avalue of unity in Sk

ti and zero outside The distribution function can then be writtenas para k

i (t) =R

laquoAgrave k

ti dx From the theory of weak convergence (see Evans 1990) thereexists a (Lebesgue measurable) density sup3 ti(x) taking values in the interval [0 1]such that limk 1 para k

i (t) =R

laquosup3 ti(x) dx The density sup3 ti is the distribution of states

of the equivalent stress brvbar ( frac14 k ) in the limit of vanishing k Here t sup3 ti(x) is theasymptotic density of states of the equivalent stress in the `homogenizedrsquo compositeAn application of theorem 42 delivers the homogenization constraints

sup3 ti(x)(f i(SE frac14 M (x)) iexcl t) gt 0 i = 1 N (112)

which hold almost everywhere on laquo It is clear that (112) is the extension of (19)to situations where the macrostress is no longer uniform

We outline the applications of the homogenization constraint (112) noting thatidentical results hold for the more general contexts of graded periodic microstructures



and microstructures specishy ed by G-converging elastic tensors The results for thegeneral cases are given in xx 3 and 4 It is clear from (112) that sup3 ti(x) = 0 on setswhere

t gt f i(SE frac14 M (x)) (113)

We choose a subset of the specimen Thus if (113) holds almost everywhere on then the measure of the sets in the ith phase where brvbar ( frac14 k ) gt t must vanish inthe shy ne-scale limit This is a special case of corollary 43 which applies to gradedperiodic composites If


for i = 1 N on then one can pass to a subsequence if necessary to shy nd foralmost every point in and for every scale k below a critical scale 0 that

brvbar ( frac14 k (x)) 6 t (115)

Here the critical scale 0 can depend on the point x in (see corollary 45) Thisshows that a pointwise bound on the macrostress modulation function delivers apointwise bound on the equivalent stress in the composite when the microstructureis sumacr ciently shy ne Next we set

~t = supx 2

(maxi

ff i(SE(A x) frac14 M (x))g)

From (115) it then follows that

brvbar ( frac14 j (x)) 6 ~t (116)

for every j lt 0 and for almost all points in As before the critical scale 0 candepend upon x This result also holds in the context of graded periodic microstruc-tures (see corollary 45) and also in the general homogenization context (see corol-lary 34)

Conversely it follows from (112) that

f i(SE frac14 M (x)) gt t (117)

on sets where sup3 ti(x) gt 0 and it is evident that f i(SE frac14 M (x)) = 1 on sets forwhich sup3 ti gt 0 for every t gt 0

We write the indicator function Agrave k

i by taking a value of unity in the ith phaseand zero outside The L 1 norm of a function g over the sample laquo is denoted bykgk 1 We consider k Agrave k

i brvbar ( frac14 k )k 1 Since Agrave k

i takes a value of unity in the ith phaseand zero outside k Agrave k

i brvbar ( frac14 k )k 1 delivers the L 1 norm of brvbar ( frac14 k ) in the ith phaseSuppose it is known that ` = lim supk 1 k Agrave k

i brvbar ( frac14 k )k 1 lt 1 Suppose further thatfor an interval 0 lt macr lt b the measure (volume) of the sets where sup3 ìexcl macr i gt 0 does notvanish We then have the upper bound given by

lim supk 1

k Agrave k

i brvbar ( frac14 k )k 1 6 kf i(SE frac14 M (x))k1 (118)

This follows in a straightforward manner from (117) see corollaries 35 and 46


926 R P Lipton

For 1 6 p lt 1 consider the Lp norm kgkp = (R

laquojg(x)jp dx)1=p Suppose that

lim supk 1 kAgrave k

i brvbar ( frac14 k )k 1 lt 1 It follows from the constraint (112) and an analysisidentical to the proof of corollary 36 that

lim supk 1

k Agrave k

i brvbar ( frac14 k )kp 6 kf i(SE frac14 M (x))k 1 j laquo j1=p (119)

for any p 1 6 p lt 1 A similar result is presented for the general case in corollary 36see also corollary 47 It turns out that the homogenization constraint obtained inthe general context of G-convergence is nearly identical to (112) except that it isgiven in terms of a sequence of periodic problems posed on the unit cube Q (seetheorem 31 and corollaries 3236)

Recent work by Pagano amp Yuan (2000) analyses the enotect of the discrete microge-ometry at the free edge of a loaded two-ply laminate Here one ply is composed of aunidirectional shy bre composite and the other is composed of homogeneous materialWhen the composite ply is modelled as a homogenized material with an enotectiveanisotropic stinotness there is a skin enotect at the free edge of the two-ply laminateHere a dramatic amplishy cation of the macrostress occurs inside a thin skin near theply boundary (see Pagano amp Rybicky 1974) Such skin enotects are found to appear inother problems of anisotropic solid mechanics (see Biot 1966) Pagano amp Yuan (2000)replace the homogenized composite near the boundary with a representative volumeelement (RVE) containing the discrete shy bre structure The RVE is subjected to themacrostress near the free edge Their numerical calculations show that the discreteshy bre phase modulates the macrostress The modulated macrostress shy eld exhibitsmuch smaller growth at the free edge It is seen that the modulated macrostressapproximates the observed behaviour of the actual stress at the free edge in the shy brereinforced composite The analysis given here shows that when the microstructure issumacr ciently shy ne relative to the ply dimensions the maximum modulated macrostressdelivers a rigorous upper bound on the actual stress near the free edge (see equations(116) (313) and (415))

It should be pointed out that one of the principal features of the theory givenhere is that the choice of RVE (given by the unit cube Q) arises naturally from thetheory and is not imposed as a hypothesis (see xx 3 and 4) Also the theory automat-ically shy xes the type of macroscopic loading for the micromechanical boundary-valueproblem posed on the RVE In the general context of G-convergence a sequence ofmicromechanical boundary-value problems associated with a sequence of microge-ometries is solved (see x 3 a)

Lastly we provide the sequence of ideas behind the homogenization constraint(112) The same set-up is used to establish the homogenization constraint in themore general settings in xx 3 and 4 One starts with the inequality

Agrave k

ti brvbar (frac14 k ) iexcl tAgrave k

ti gt 0 (120)

This follows from the deshy nition of the set Sk

ti The method developed here treats theset Sk

ti as a new material phase with material property PN + 1 The piecewise constantelasticity tensor for this new composite is written as Ck (A1 AN PN + 1 x)The elasticity tensor for the actual composite is recovered by setting PN + 1 = Ai

in Sk

ti Because the sets Sk

ti are not k-periodic one cannot use periodic homoge-nization but instead must work in the general homogenization context provided bythe G-convergence (see Dal Maso 1993) The enotective elasticity or G-limit for the



sequence fCk (A1 AN PN + 1 x)g 1k = 1 is denoted by eCE(A1 PN + 1 x) The

distributional limit of the sequence of products fAgrave k

ti brvbar ( frac14 k )g 1k = 1 as k tends to zero

is identishy ed as

(AibrvbarAirN + 1 eCE(A1 AN Ai x))SE frac14 M (x) SE frac14 M (x) (121)

(see theorem 23) Here rN + 1 eCE(A1 AN Ai x) is the gradient of the G-limiteCE with respect to the variable PN + 1 evaluated at PN + 1 = Ai It is shown in x 4that this limit is bounded above by sup3 ti(x)f i(SE frac14 M (x)) for almost every x in thespecimen This bound follows from new explicit local formulae for the gradients ofG-limits given in theorem 22 (see also Lipton 2003) The constraint (112) thenfollows by passing to the limit in the sense of distributions in (120) and applyingthe bound to the distributional limit of fAgrave k

ti brvbar ( frac14 k )g 1k = 1 described above In x 4

inequalities analogous to (112) are established for the more general case of gradedperiodic microstructures in theorem 42 In x 3 minimal hypotheses are placed on themicrostructure and an inequality analogous to (112) is established in theorem 31The minimal hypothesis is that the sequence of elastic tensors describing the micro-geometry G converge as the scale of the microstructure tends to zero It is pointedout that this hypothesis is weak in the sense that any sequence of conshy gurationswith elastic tensors taking values Ai in the ith material is guaranteed to have aG-converging subsequence (see Spagnolo 1976 Murat amp Tartar 1997 Simon 1979)

In order to expedite the presentation the following notation is used The con-tractions of symmetric fourth-order tensors A with second-order tensors frac14 M arewritten [A]ijkl frac14

Mkl = A frac14 M contractions of two fourth-order tensors A and M are

written [A]ijkl[M ]klop = AM contractions of two second-order tensors frac14 deg are writ-ten frac14 ij deg ij = frac14 deg frac14 frac14 = j frac14 j2 and contractions of second-order tensors deg and vectorsx are written deg ijxj = deg x The Lebesgue measure of a set S is denoted by jSj

2 Homogenization of mean-square stress deguctuations andlower bounds on stress concentrations

We consider graded composite materials made from N linearly elastic phases Theelastic phases occupy the N subsets laquo 1 laquo 2 laquo N where laquo 1 [ laquo 2 [ cent cent cent [ laquo N =laquo Here we make minimal assumptions on the shapes of the subsets and simplyassume that they are Lebesgue measurable The characteristic length-scale of themicrogeometry relative to the size of the structure is denoted by The local elasticitytensor C(A x) is piecewise constant and takes the value C(A x) = Ai in the ithphase Denoting the indicator function of the ith material by Agrave

i the local elasticitytensor is written C(A x) =

PNi= 1 Agrave

i Ai Here Agrave i takes a value of unity for points

inside the ith phase and zero outsideFor a body load f the elastic displacement u is the solution of

iexcl div(C(A x) deg (u(x))) = f (21)

At each point in the composite the stress frac14 is given by the constitutive relation

frac14 (x) = C(A x) deg (u(x)) (22)

Here deg (u) is the strain tensor given by

deg (u) = 12(u

ij + uji) (23)


928 R P Lipton

where ui is the ith component of displacement u and u

ij is its partial derivativein the jth coordinate direction The prescribed displacement on the boundary ofthe structure is zero The solution u is square integrable and has square-integrablederivatives The space of such functions that vanish on the boundary is denotedby H1

0 ( laquo )3 The dual space is written as Hiexcl1( laquo )3 and the body force f is anelement of this space (see Duvaut amp Lions 1976) It is assumed that the sequence ofelastic tensors fC(A x)ggt0 G-converge to the enotective elastic tensor CE(A x)The deshy nition of G-convergence is given below (see Spagnolo 1976)

Demacrnition 21 (G-convergence) The sequence fC(A x)ggt0 is said to G-converge to CE if for every load f (in Hiexcl1( laquo )3) the sequence of elastic displace-ments fuggt0 converges weakly (in H1

0 ( laquo )3) to u M where

iexcl div(CE(A x) deg (uM (x))) = f (24)

The enotective tensor or G-limit CE(A x) depends upon x and is a measurable func-tion (see Spagnolo 1976) Here the sequence of stresses f frac14 ggt0 converges weakly (inL2( laquo )3pound3) to the macroscopic stress frac14 M and the macroscopic constitutive relationis given by

frac14 M (x) = CE(A x) deg (u M (x)) (25)

The hypotheses necessary for G-convergence are minimal Indeed for any sequencefC(A x)ggt0 described above one is guaranteed the existence of a G-convergentsubsequence (see Murat amp Tartar 1997 Simon 1979 Spagnolo 1976)

We start by computing the limit as tends to zero of Agrave i brvbar ( frac14 ) in the sense of

distributions We show that this limit is given in terms of derivatives of the enotec-tive tensor CE(A x) In order to deshy ne derivatives of G-limits we introduce theneighbourhood N (A) of the array A Arrays in this neighbourhood are denoted byP = (P1 P2 PN ) The neighbourhood is chosen such that all tensors Pi in thearray satisfy the constraint 0 lt para lt Pi lt curren Here para and curren are bounds on the smallestand largest eigenvalues of the elasticity tensors

The elasticity tensor associated with a particular choice of component elastic ten-sors is written

C(P x) =NX

i= 1

Agrave i Pi and C(A x) =

NX

i = 1

Agrave i Ai

Passing to a subsequence if necessary one is guaranteed that the G-limit CE(P x)of fC(P x)ggt0 exists for every P in N (A) (see Tartar 2000)

We introduce a sequence of cubes Q(x r) of diameter r containing a prescribedpoint x in laquo For r sumacr ciently small Q(x r) is contained within laquo Given a constantstrain middotdeg applied to the cube the local oscillatory response function is denoted by wr

middotdeg where wr

middotdeg is the Q(x r)-periodic solution of

iexcl div(C(A y)( deg (wrmiddotdeg (y)) + middotdeg )) = 0 for y in Q(x r) (26)

The local response functions are used to deshy ne directional derivatives of the G-limitin the following theorem

Theorem 22 The directional derivative of CE(P x) at P = A with respect tothe ith component elasticity in the direction specimacred by the symmetric fourth-order



tensor Mi is given by

CE(A x)

Mimiddotdeg middotdeg = MiriCE(A x)middotdeg middotdeg

=3X

k = 1

3X

l = 1

3X

m= 1

3X

n= 1

[Mi]klmn limr 0

lim 0

micro1

jQ(x r)j

para

poundZ

Q(x r)

Agrave i ( deg (wr

middotdeg )kl + middotdeg kl)( deg (wrmiddotdeg )mn + middotdeg mn) dy (27)

where middotdeg is any constant strain The derivative is Lebesgue measurable with respectto x

Here the ith phase gradient riklmnCE(A x)middotdeg middotdeg is given by the local formulae

riklmnCE(A x)middotdeg middotdeg

= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr


and

riklklC

E(A x)middotdeg middotdeg

= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr

middotdeg )kl + middotdeg kl)( deg (wrmiddotdeg )kl + middotdeg kl) dy (29)

where repeated indices indicate summation The formula for the G-limit is

CE(A x)middotdeg middotdeg = limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

C(A y)( deg (wrmiddotdeg ) +middotdeg ) ( deg (wr

middotdeg ) + middotdeg ) dy

(210)These formulae hold for any sequence of cubes Q(x r) that contain x for everyr gt 0 The local formula (210) for the G-limit was developed earlier (see Spagnolo1976) The formulae (27) and (28) for the derivatives of the enotective elastic tensorare semi-explicit in that they are given in terms of sequences of solutions to localproblems These formulae hold for general oscillations and are obtained without anyhypotheses on the sequence of conshy gurations

The distributional limit of Agrave i brvbar ( frac14 ) is given in the following theorem

Theorem 23 (homogenization of stress deguctuations in the ith phase)Given that fC(A x)ggt0 G-converges to CE(A x) as tends to zero

Agrave i brvbar ( frac14 (x)) brvbarHi(x) frac14 M (x) frac14 M (x) (211)

in the sense of distributions and

brvbarHi(x) frac14 M (x) frac14 M (x) = (SE(A x)AibrvbarAiriCE(A x)SE(A x)) frac14 M (x) frac14 M (x)(212)

Here SE(A x) = (CE)iexcl1(A x) is the eregective compliance


930 R P Lipton

Passing to a subsequence if necessary one shy nds that f Agrave i ggt0 converges weakly to

a density sup3 i(x) such that for any cube Q(x r) in laquo one has

lim 0

Z

Q(x r)

( Agrave i (y) iexcl sup3 i(y)) dy = 0

(see Evans 1990) This convergence is written as Agrave i

curren sup3 i in L1 ( laquo )

The next result gives a lower bound on the stress concentrations in the limit = 0

Theorem 24 Given that fC(A x)ggt0 G-converges to CE(A x) Agrave i

curren sup3 i

brvbarHi(x) frac14 M (x) frac14 M (x) 6 sup3 i(x) lim inf 0

k Agrave i brvbar (frac14 )k 1 (213)

almost everywhere One also has the following theorem

Theorem 25 Given that fC(A x)ggt0 G-converges to CE(A x)degdegdegdeg

NX

i= 1

brvbarHi frac14 M frac14 M

degdegdegdeg1

6 lim inf 0

k brvbar ( frac14 )k1 (214)

An illustrative example of the lower bound is given for a long shaft subjectedto torsion loading Here the cross-section of the shaft is composed of a periodicchequerboard pattern of two materials of shear stinotness G1 and G2 and we choosebrvbar ( frac14 ) = j frac14 j2 For torsion loading the non-vanishing components of stress are frac14

13frac14

23 One has2X

i = 1

brvbarHi =G1 + G2

2p

G1G2

(215)

andG1 + G2

2p

G1G2

k( frac14 M13)2 + ( frac14 M

23)2k 1 6 lim inf 0

k( frac14 13)2 + ( frac14

23)2k 1 (216)

It is clear from this example that the enotect of the tensorP2

i = 1 brvbarHi is signishy cantas it diverges when either shear modulus tends to zero

The proofs of theorems 2225 are given in the following subsection

(a) Proofs of theorems 2225

First theorem 22 is established Consider a sequence fC(P x)ggt0 that G-converges to CE(P x) for every P in N (A) Given a constant strain middotdeg the localoscillatory response wr

middotdeg is the Q(x r) periodic solution of (26) The dinoterentialequation (26) is written in the weak form given by

Z

Q(xr)

C(A y)( deg (wrmiddotdeg ) + middotdeg ) deg (v) dy = 0 (217)

for all Q(x r) periodic v that are square integrable and have square-integrable partialderivatives The volume of Q(x r) is denoted by jQ(x r)j and an application ofCauchyrsquos inequality in (217) gives

Z

Q(xr)

j deg (wrmiddotdeg )j2 dy 6 ( curren 2=para 2)jQ(x r)jjmiddotdeg j2 (218)



and Z

Q(xr)

j deg (wnmiddotdeg ) + middotdeg j2 dy 6 ( curren 2=para 2 + 1)jQ(x r)jjmiddotdeg j2 (219)

Choose a number macr shy and M such that A + macr shy M are in N (A) and consider thesequence fC(A + macr shy M x)ggt0 Here M is an array of elastic tensors that is iden-tically zero except for the ith component elastic tensor Mi Here Mi is of norm onewhere the tensor norm is given by

jMij =

s X

klmn

[Mi]2klmn

The sequence of coemacr cients fC(A + macr shy M x)ggt0 dinoter from fC(A x)ggt0 bythe increment macr shy Mi Agrave

i The G-limit for the sequence fC(A + macr shy M x)ggt0 is writ-

ten as CE(A + macr shy M x) We set macr C = CE(A + macr shy M x) iexcl CE(A x) and use (210)to compute macr C with respect to the increment macr shy The oscillatory responses associ-ated with the sequence fC(A + macr shy M x)ggt0 are denoted by wr

middotdeg where wrmiddotdeg are

Q(x r)-periodic solutions ofZ

Q(xr)

C(A + macr shy M y)( deg (wrmiddotdeg ) + middotdeg ) deg (v) dy = 0 (220)

for all Q(x r)-periodic v that are square integrable and have square-integrablederivatives

It follows from (210) that

macr Cmiddotdeg middotdeg

= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)

C(A + macr shy M y)( deg (wrmiddotdeg ) + middotdeg ) ( deg (wr


iexcl limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

C(A y)( deg (wrmiddotdeg ) + middotdeg ) (wr


(221)

Writing

C(A + macr shy M y) = C(A y) + macr shy Mi Agravei and wr

middotdeg = wrmiddotdeg + macr wr

where macr wr = wrmiddotdeg iexcl wr

middotdeg one has for every Q-periodic v the equation

0 =

Z

Q(x r)

macr shy Mi Agravei ( deg (wr

middotdeg ) + middotdeg ) deg (v) dy

+

Z

Q(x r)

C(A + macr shy M y) deg ( macr wr) deg (v) dy (222)

Setting v = macr wr in (222) and substitution into (221) gives

macr Cmiddotdeg middotdeg = macr shy limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Mi Agravei ( deg (wr

middotdeg ) + middotdeg ) ( deg (wrmiddotdeg ) + middotdeg ) dy

+ macr shy limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)


middotdeg ) + middotdeg ) deg ( macr wr) dy

(223)


932 R P Lipton

Setting v = macr wr in (222) and application of Cauchyrsquos inequality givesZ

Q(xr)

j deg ( macr wr)j2 dy 6 macr shy 2 para iexcl2jQ(x r)jjmiddotdeg j2(1 + ( curren =para )2) (224)

Estimates (219) and (224) show thatmacrmacrmicro

1

jQ(x r)j

para Z

Q(xr)



macrmacr 6 macr shy jmiddotdeg j2(1 + ( curren =para )2) (225)

From this one deduces that


lim 0

micro1

jQ(x r)j

para Z

Q(x r)


middotdeg ) +middotdeg ) ( deg (wrmiddotdeg ) +middotdeg ) dy +o( macr shy )

(226)and (27) follows The measurability of CE(A x)=Mi is assured as it is a pointwiselimit of measurable functions and theorem 22 is proven

Theorem 23 follows from arguments analogous to those presented in Lipton (20022003)

To establish theorem 24 note that for any positive test function p in C 1 ( middotlaquo )Holderrsquos inequality gives

Z

laquo

p(x)( Agrave i brvbar ( frac14 ) iexcl Agrave

i kAgrave i brvbar ( frac14 )k 1 ) dx 6 0 (227)

Sending to zero together with theorem 23 givesZ

laquo

p(x)sup3

brvbarHi(x) frac14 M (x) frac14 M (x) iexcl sup3 i(x) lim inf 0

k Agrave i brvbar ( frac14 )k1

acutedx 6 0 (228)

for every choice of p and theorem 24 follows Theorem 25 also follows from Holderrsquosinequality

3 A homogenized constraint condition and local bounds on stress

In this section homogenized quantities are introduced that provide information aboutthe size of sets where the actual stress is large We consider a sequence of elastictensors fCj (A x)g1

j = 0 where the size of the microgeometry relative to the domainis given by j j = 1 2 and j tends to zero as j goes to inshy nity Here

Cj (A x) =

NX

i= 1

Agravej

i Ai

where Agravej

i is the characteristic function of the ith phase We suppose that thesequence fCj (A x)g 1

j = 0 G-converges to the enotective elastic tensor CE(A x) Abody load f is applied to the domain laquo and the elastic displacements uj are solu-tions of

iexcl div(Cj (A x) deg (uj (x))) = f (31)

with uj = 0 on the boundary of laquo The actual stress in the composite satisshy esfrac14 j = Cj (A x) deg (uj (x))



The stresses f frac14 j g 1j = 1 converge weakly in L2( laquo )3pound3 to the homogenized stress frac14 M

given by frac14 M = CE(A x) deg (u M ) and the displacements fuj g 1j = 1 converge weakly to

uM (in H10 ( laquo )3) Here u M is the solution of the homogenized problem given by

iexcl div(CE(A x) deg (uM )) = f (32)

and u M = 0 on the boundary of laquo The enotective compliance is given by (CE(A x))iexcl1

and is denoted by SE(A x)We track the behaviour of the local stresses in each phase in the shy ne-scale limit

Consider the set Sj

ti of points in the ith phase where brvbar ( frac14 j ) gt t Choose a subset of the sample Passing to a subsequence if necessary one has a density sup3 ti(x) takingvalues in the unit interval for which

limj 1

jSj

ti j =

Z

sup3 ti dx (33)

Here sup3 ti gives the asymptotic distribution of states for the equivalent stress brvbar ( frac14 j ) inthe limit of vanishing j It is evident that if sup3 ti = 0 on then the volume (measure)of the sets in where the equivalent stress exceeds t in the ith phase vanishes withj We provide a link between sup3 ti and the macroscopic stress frac14 M appearing in thehomogenized composite This link is given by the macrostress modulation function Inthis section the function is deshy ned in the context of G-convergence For the simplercase of periodic homogenization the macrostress modulation function is given byequation (15)

Consider a sequence of cubes Q(x rj) centred at a point x in laquo together with thevalues of Cj (A y) for points y in Q(x rj) It is assumed that rj 0 as j goes toinshy nity and j=rj 0 On re-scaling one considers Cj (A x + rjz) for z in the unitcube Q centred at the origin and

Cj (A x + rjz) =

NX

i = 1

Agravej

i (x + rjz)Ai

We introduce the tensor F ij (x z) deshy ned on laquo pound Q such that for every constant

symmetric 3 pound 3 tensor middotdeg it is given by

F ij (x z)middotdeg middotdeg = Agrave

j

i (x + rjz)brvbarAi( deg z (wj(x z)) + middotdeg ) Ai( deg z (wj(x z)) + middotdeg )

= Agravej

i (x + rjz) brvbar (Ai( deg z (wj(x z)) + middotdeg )) (34)

where for each x in laquo the function wj(x z) is the Q-periodic solution of

iexcl divz (Cj (A x + rjz)( deg z (wj(x z)) + middotdeg )) = 0 for z in Q (35)

Here x appears as a parameter and dinoterentiation is carried out in the z variable asindicated by the subscripts For shy xed x the L 1 (Q) norm of

Agravej

i (x + rjz) brvbar (Ai( deg z (wj(x z)) + middotdeg ))

is computed This is written as

kF ij (x cent)SE(A x) frac14 M (x) SE(A x) frac14 M (x)kL 1 (Q) (36)

Denote by C the class of all sequences frjg 1j = 1 such that

limj 1

rj = 0 and limj 1

j=rj = 0


934 R P Lipton

Set

~f i(x SE(A x) frac14 M (x))

= supfrj g1

j=12 C

nlim sup

j 1kF i

j (x cent)SE(A x) frac14 M (x) SE(A x) frac14 M (x)kL 1 (Q)

o (37)

The function ~f i(x SE(A x) frac14 M (x)) is the macrostress modulation function We nowhave the following theorem

Theorem 31 (the homogenization constraint) If we consider a sequencefCj (A x)g 1

j = 1 with the associated 2N + 1-tuple

(CE(A x) sup3 t1(x) sup3 tN (x) ~f1(x SE(A x) frac14 M (x)) ~f N(x SE(A x) frac14 M (x)))

then we have

sup3 ti(x)( ~f i(x SE(A x) frac14 M (x)) iexcl t) gt 0 i = 1 N (38)

almost everywhere on laquo

The next corollary is an immediate consequence of theorem 31

Corollary 32 Given the hypothesis of theorem 31 suppose that

t gt ~f i(x SE(A x) frac14 M (x)) (39)

on a set where raquo laquo Then the stress in the ith phase satismacres brvbar ( frac14 j ) 6 t almosteverywhere on outside S

j

ti moreover the measure of the set Sj

ti tends tozero in the macrne-scale limit

We denote the set of points in laquo where brvbar ( frac14 j ) gt t by Sj

t and one has thefollowing corollary



for i = 1 N on a set where raquo laquo Then brvbar ( frac14 j ) 6 t almost everywhere on outside S

j

t moreover the measure of the set Sj

t tends to zero in themacrne-scale limit

We demonstrate that a pointwise bound on the macrostress modulation func-tion delivers a pointwise bound on the equivalent stress in the composite when themicrostructure is sumacr ciently shy ne relative to the size of the domain and variation ofthe body force

Corollary 34 (pointwise bounds on the equivalent stress) Given thehypothesis of theorem 31 suppose that


for i = 1 N on a set where raquo laquo One can then pass to a subsequence ifnecessary to macrnd that there is a critical 0 such that for every j lt 0

brvbar ( frac14 j (x)) 6 t (312)

for almost every x in Here 0 can depend upon position



Set~t = sup

x 2

sup3max

if ~f i(x SE(A x) frac14 M (x))g

acute

From corollary 34 one then has a critical scale 0 (depending possibly on x) suchthat for every j lt 0

brvbar ( frac14 j (x)) 6 ~t (313)

for almost all points in

Proof of corollary 34 Since limj 1 jSj

t j = 0 we can choose

fCjk (A x )g 1k = 1

with stresses f frac14 jk g1k = 1 that satisfy jSjk

t j lt 2iexclk Then if x does not belong toS 1kgt ~K S

jkt one has that brvbar ( frac14 jk ) 6 t for every k gt ~K Hence for any x not

in A =T 1

K = 1

S 1kgt K S

jkt there is an index K for which brvbar ( frac14 jk ) 6 t for every

k gt K But

jAj 6macrmacr

1[

kgt ~K

Sjkt

macrmacr 6

1X

k = ~K

jSjkt j 6 2iexcl ~K + 1

Hence jAj = 0 Thus for almost every x in there is a shy nite index K (that maydepend upon x) for which brvbar ( frac14 jk ) 6 t for every k gt K and the corollary follows yen


~f i(x SE(A x) frac14 M (x)) gt t (314)

on sets where sup3 ti(x) gt 0 It is evident that ~f i(x SE(A x) frac14 M (x)) = 1 on sets forwhich sup3 ti gt 0 for every t gt 0

Next we provide conditions for which upper bounds on lim supj 1 k Agravedeg j

i brvbar ( frac14 deg j )k 1can be obtained in terms of homogenized quantities

Corollary 35 Suppose it is known that ` = lim supj 1 kAgravedeg j

i brvbar ( frac14 deg j )k 1 lt 1Suppose further that for an interval 0 lt macr lt b the measure (volume) of the setswhere sup3 ìexcl macr i gt 0 does not vanish Then

lim supj 1

kAgravedeg j

i brvbar ( frac14 deg j )k 1 6 k ~f i(x SE(A x) frac14 M (x))k 1 (315)

Corollary 35 follows upon setting t = ` iexcl macr and applying (314)Next bounds are obtained on lim supj 1 kAgrave

deg j

i brvbar ( frac14 deg j )kp for 1 6 p lt 1

Corollary 36 Suppose that lim supj 1 kAgravedeg j

i brvbar ( frac14 deg j )k 1 lt 1 Then

lim supj 1

k Agravedeg j

i brvbar ( frac14 deg j )kp 6 k ~f i(x SE(A x) frac14 M (x))k 1 j laquo j1=p

for any p 1 6 p lt 1

Proof of corollary 36 For t gt 0 we consider the set of points Sj

ti in the ithphase where brvbar ( frac14 j ) gt t The distribution function para j

i (t) is deshy ned by para ji (t) = jSj

tijFor each j the distribution function is non-increasing continuous from the right and


936 R P Lipton

therefore measurable Set ` = lim supj 1 k Agravedeg j

i brvbar ( frac14 deg j )k 1 Choose r gt 0 then fromthe hypotheses para j

i has support in [0 ` + r) for sumacr ciently large j For each j one hasZ

laquo

Agravej

i ( brvbar ( frac14 deg j ))p dx = p

Z 1

0

tpiexcl1 para ji (t) dt (316)

It is easy to see that the function gp(t) deshy ned by gp(t) = tpiexcl1j laquo j for 0 6 t lt ` + rand gp(t) = 0 for ` + r 6 t lt 1 is integrable on [0 1) and tpiexcl1 para j

i (t) 6 gp(t) forsumacr ciently large j Thus an application of Lebesguersquos theorem gives

lim supj 1

Z

laquo

Agravej

i (brvbar ( frac14 deg j ))p dx 6 p

Z 1

0

tpiexcl1 lim supj 1

( para ji (t)) dt (317)

For each t we pass to a subsequence if necessary and appeal to (33) to obtain adensity sup3 ti(x) for which

lim supj 1

para ji (t) =

Z

laquo

sup3 ti(x) dx

with 0 6 sup3 ti(x) 6 1 It follows from theorem 31 that sup3 ti(x) = 0 almost everywhereif t gt ~J = k ~f i(x SE(A x) frac14 M (x))k 1 It is now evident that

lim supj 1

Z

laquo

Agravej

i ( brvbar ( frac14 deg j ))p dx 6 p

Z ~J

0

tpiexcl1

Z

laquo

sup3 ti(x) dx dt (318)

The corollary follows from the estimate j laquo j gtR

laquosup3 ti(x) dx yen

(a) Proof of theorem 31

Consider the set of points Sj

ti in the ith phase for which brvbar ( frac14 j ) gt t for a shy xedpositive number t The characteristic function for S

j

ti is denoted by Agrave jti(x) Passing

to a subsequence if necessary one has that Agrave jti

curren sup3 ti in L1 ( laquo ) as j 1 Here the

density sup3 ti(x) takes values in [0 1] For any non-negative smooth test function p(x)one has

t

Z

laquo

pAgrave jti dx 6

Z

laquo

pAgrave jti brvbar ( frac14 j ) dx (319)

Taking the limit on the left-hand side of (319) gives tR

laquopsup3 ti dx The goal is to pass

to the limit on the right-hand side of (319) to obtain a pointwise upper bound onthe product tsup3 ti This will follow from the following lemma

Lemma 37 There exists a subsequence of f Agrave jtig 1

j = 1 and f frac14 j g 1j = 1 also denoted

by fAgrave jtig1

j = 1 and f frac14 j g 1j = 1 such that

Agrave jti brvbar ( frac14 j ) AibrvbarAiHt(x)SE(A x) frac14 M (x) SE(A x) frac14 M (x)

in the sense of distributions Here AibrvbarAiHt(x) is given by

AibrvbarAiHt(x)middotdeg middotdeg

= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Agrave jtibrvbar(Ai( deg (w

jrmiddotdeg ) + middotdeg )) (Ai( deg (w

jrmiddotdeg ) + middotdeg )) dy

= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Agrave jti brvbar (Ai( deg (w

jrmiddotdeg ) + middotdeg )) dy (320)



where middotdeg is any constant strain and the local oscillatory response function is denotedby wjr

middotdeg Here wjrmiddotdeg is the Q(x r)-periodic solution of

iexcl div(Cj (A y)( deg (wjrmiddotdeg (y)) + middotdeg )) = 0 for y in Q(x r) (321)

Proof of lemma 37 One introduces a sequence of N + 1 phase composites withthe following properties on the set S

j

ti the elasticity is PN + 1 for points in the ithphase not in S

j

ti the elasticity is Pi in the other phases the elasticity is Pm in themth phase m = 1 N for m 6= i We denote the elastic tensor for this compositeby Cj (P PN + 1 x) This tensor is deshy ned in terms of characteristic functions by

Cj (P PN + 1 x) =

NX

m= 1m 6= i

Agrave jm(x)Pm + ( Agrave

j

i (x) iexcl Agrave jti(x))Pi + Agrave j

ti(x)PN + 1

Most importantly we observe that Cj (A x) = Cj (A Ai x) Passing to sub-sequences as necessary the sequence

fCj (P PN + 1 x)g1j = 1

G-converges to CE(P PN + 1 x) for every (P PN + 1) in a neighbourhood of (A Ai)Observing that AibrvbarAiHt(x) = AibrvbarAirN + 1CE(A Ai x) the lemma then followsfrom the same arguments as those used to prove theorem 23 yen

We pass to subsequences according to lemma 37 and on taking limits in (319) itfollows that

AibrvbarAiHt(x)SE(A x) frac14 M (x) SE(A x) frac14 M (x) iexcl tsup3 ti(x) gt 0 (322)

almost everywhere on laquo Passing to a diagonal sequence we have limj 1 rj = 0limj 1 j=rj = 0

AibrvbarAiHt(x)middotdeg middotdeg = limj 1

micro1

jQ(x rj)j

para Z

Q(x rj )


jrj

middotdeg ) + middotdeg )) dy (323)

and at Lebesgue points of sup3 ti

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Agrave jti dx = sup3 ti(x) (324)

Here wjrj

middotdeg solves (321) with r = rj Now change coordinates by translation andre-scaling so that Q(x rj) is transformed to the unit cube Q centred at the originThe change of variables is given by y = x + zrj where z lies in Q One writesw

jrj

middotdeg (y) = rj iquest j((y iexcl x)=rj) where iquest j(z) is the Q-periodic solution of

iexcl div(Cj (A x + rjz)(deg ( iquest j(z)) + middotdeg )) = 0 for z in Q (325)

Under this change of coordinates Agrave jti = Agrave j

ti(x + rjz) and


Z

Q

Agrave jti(x + rjz) brvbar (Ai( deg ( iquest

j(z)) + middotdeg )) dz (326)


938 R P Lipton

Application of Holderrsquos inequality gives


6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x + rjz) dz

paralim sup

j 1kF i

j (x cent)middotdeg middotdeg kL 1 (Q)

= sup3 ti(x) lim supj 1

kF ij (x cent)middotdeg middotdeg kL 1 (Q) (327)

Theorem 31 now follows from (322) and noting that the upper bound (327) holdsfor a subsequence of f Agrave

j

i g 1j = 1 and for a particular choice of frjg 1

j = 1 in C

4 A homogenized constraint condition and bounds on stress forpiecewise constant and continuously graded microstructures

In this section we consider microstructures that are piecewise constant or changecontinuously across the structure Here the microstructure is assumed to be locallyperiodic A graded locally periodic composite requires a description at both macro-scopic and microscopic length-scales We start by describing the periodic microstruc-ture Periodic conshy gurations of N elastic phases in the unit cube Q are consid-ered Here Q is the unit cell for the periodic microstructure however more generalperiod cells could be used The elasticity tensor of each phase is specishy ed by Ai fori = 1 N A particular choice of component elasticity tensors is specishy ed by thearray A = (A1 A2 A3 AN ) The elasticity tensor for a given conshy guration inthe unit cube is written C0(A y) Here C0(A y) = Ai for y in the ith phase and isperiodic in y

A periodic microstructure occupying laquo of length-scale 1=cedil is described by thesimple re-scaling C0(A cedil x) Here the periodic microgeometry remains the same asx traverses laquo To describe the spatial variation of the microstructure across laquo itis convenient to start with a function C0(A x y) deshy ned on laquo pound Q For each xthe function C0(A x cent) describes an elasticity tensor for a conshy guration in the unitperiod cell Q We consider two types of graded locally periodic composites Theshy rst locally periodic microstructure corresponds to periodic microgeometries thatare constant on shy xed subdomains of laquo The structural domain laquo is partitioned intosubdomains O micro

` of diameter less than or equal to micro Here laquo =S

` O micro` Associated

with each set O micro` we consider a function C`

0(A y) The piecewise constant locallyperiodic microgeometry is given in terms of the function

C0(A x y) = C`0(A y) (41)

when x lies in O micro` For cedil = 1 2 the locally periodic microstructure is given

by C0(A x x ) Here the microgeometry is the same inside each O micro` but can be

dinoterent for each O micro` The shy ne-scale limit for the family of composites is obtained by

sending 1=cedil to zero This type of graded microstructure is considered in Bensoussanet al (1978) The enotective elastic property obtained in the shy ne-scale limit is derivedin Bensoussan et al (1978 ch 6) Here the enotective elastic tensor is a function ofposition taking dinoterent values on each set O micro

` This type of microgeometry is referredto in this article as a piecewise constant locally periodic microstructure

The second type of graded composite admits a continuous variation of microscopicproperties One considers functions C0(A x y) that are continuous in the x variable



Here the continuity is specishy ed by

limh 0

Z

Q

jC0(A x + h y) iexcl C0(A x y)j dy = 0 (42)

As before the structural domain laquo is partitioned into subdomains O micro` of diameter

less than or equal to micro The locally periodic microgeometry is given in terms ofthe function denoted by C micro

0 (A x y) and is deshy ned in the following way in the `thsubdomain C micro

0 (A x y) = C0(A x` y) where x` is a sample point chosen from O micro`

The locally periodic microstructure is described by the elasticity tensor C micro0 (A x cedil x)

The continuity condition (42) ensures that near by subdomains have nearly the samemicrogeometies This type of microstructure is referred to as a continuously gradedlocally periodic microstructure In this context the shy ne-scale limit is obtained byconsidering a family of partitions indexed by j = 1 2 with subdomains O

micro j

` ofdiameter less than or equal to microj Here microj goes to zero as j goes to inshy nity and thescale of the microstructure is given by 1=cedil j where 1=cedil j goes to zero as j tends toinshy nity and (1=cedil j)=microj 0 The associated sequence of elasticity tensors is writtenas fC

micro j

0 (A x x j)g 1j = 1

To expedite the presentation we introduce the functions Agrave i(x y) i = 1 N deshy ned on laquo pound Q such that for each x the function Agrave i(x cent) is the characteristicfunction of the ith phase in the unit period cell Q For x shy xed Agrave i(x y) = 1 when yis in the ith phase and zero otherwise and we write

C0(A x y) =

NX

i

Agrave i(x y)Ai (43)

The enotective elastic tensor is deshy ned by

CE(A x)middotdeg middotdeg =

Z

Q

C0(A x y)(deg (w) + middotdeg ) ( deg (w) + middotdeg ) dy (44)

where for each shy xed value of x the function w(x y) is the Q-periodic solution of

iexcl divy (C0(A x y)(deg y (w(x y)) + middotdeg )) = 0 (45)

Here x appears as a parameter and the subscripts indicate dinoterentiation with respectto the y variable

The enotective property associated with the shy ne-scale limit of the piecewise constantperiodic microstructure is given by (44) with C0(A x y) given by (41) this isproved in Bensoussan et al (1978)

Next we consider continuously graded locally periodic microstructures Thehomogenization limit of the elasticity tensors fC

micro j

0 (A x x j)g 1j = 1 is given by the

following theorem

Theorem 41 The sequence of local elasticity tensors fCmicro j

0 (A x x cedil j)g 1j = 1 is a

G-convergent sequence and its G-limit is the eregective tensor CE(A x) demacrned by(44) Conversely for any eregective tensor CE(A x) given by (44) where C0(A x y)is given by (43) and satismacres (42) one can construct a sequence fC

micro j

0 (A x x cedil j)g1j = 1

that G-converges to it The eregective tensor is a continuous function of position


940 R P Lipton

In what follows we consider the shy ne-scale limits of piecewise constant locallyperiodic microstructures and continuously graded locally periodic microstructuresBecause there are more technicalities involved for continuously graded locally peri-odic microstructures we state and prove the following theorems for this case andnote that the same holds for the piecewise constant locally periodic microstructures

We prescribe a function of the form C0(A x y) given by (43) and satisfying(42) We consider the associated sequence of elasticity tensors fC

micro j

0 (A x x cedil j)g 1j = 1

corresponding to a sequence of continuously graded microstructures For a prescribedbody load f the elastic displacements u cedil j are solutions of

iexcl div(Cmicro j

0 (A x x cedil j)deg (u cedil j (x))) = f (46)

with u cedil j = 0 on the boundary of laquo The associated stresses are given by frac14 cedil j =C

micro j

0 (A x x j) deg (u cedil j (x))From theorem 41 the sequence fC

micro j

0 (A x x cedil j)g 1j = 1 G-converges to the enotective

elasticity tensor CE(A x) given by (44) The stresses f frac14 cedil j g 1j = 1 converge weakly

in L2( laquo )3pound3 to the homogenized stress frac14 M given by frac14 M = CE(A x)deg (u M ) and thedisplacements fucedil j g 1

j = 1 converge weakly to uM in H10 ( laquo )3 Here u M is the solution

of the homogenized problem given by

iexcl div(CE(A x) deg (u M )) = f (47)

and uM = 0 on the boundary of laquo The enotective compliance is given by (CE(A x))iexcl1


Consider the set Scedil j

ti of points in the ith phase where brvbar (frac14 cedil j ) gt t Passing to asubsequence if necessary one has the asymptotic density of states sup3 ti(x) takingvalues in the unit interval Following x 3 we provide a link between sup3 ti and themacroscopic stress frac14 M appearing in the homogenized composite We introduce thetensor F i(x y) deshy ned for every constant symmetric 3 pound 3 tensor middotdeg by

F i(x y)middotdeg middotdeg = Agrave i(x y) brvbar (Ai( deg y (w(x y)) + middotdeg )) (48)

where for each x in laquo the function w(x y) is the Q-periodic solution of (45) Themicrostress modulation function is given by

f i(x SE(A x) frac14 M (x)) = kF i(x cent)SE(A x) frac14 M (x) SE(A x) frac14 M (x)kL 1 (Q) (49)

We then have the following theorem

Theorem 42 (the homogenization constraint) If we consider a sequencefC

micro j

0 (A x x j)g 1j = 1 with the associated 2N + 1-tuple

(CE(A x) sup3 t1(x) sup3 tN (x)f1(x SE(A x) frac14 M (x)) f N(x SE(A x) frac14 M (x)))

then we have

sup3 ti(x)(f i(x SE(A x) frac14 M (x)) iexcl t) gt 0 i = 1 N (410)






t gt f i(x SE(A x) frac14 M (x)) (411)

on a set where raquo laquo The stress in the ith phase then satismacres brvbar ( frac14 cedil j ) 6 t almosteverywhere on outside S

cedil j

ti Moreover the measure of the set Scedil j


We denote the set of points in laquo where brvbar ( frac14 cedil j ) gt t by Scedil j




for i = 1 N on a set where raquo laquo Then brvbar ( frac14 cedil j ) 6 t almost everywhere on outside S

cedil j

t moreover the measure of the set Scedil j


As in the general case one shy nds that a pointwise bound on the macrostress mod-ulation function delivers a pointwise bound on the equivalent stress in the compositewhen the microstructure is sumacr ciently shy ne relative to the size of the domain andvariation of the body force



for i = 1 N on a set where raquo laquo Then one can pass to a subsequence ifnecessary to macrnd that there is a critical 0 such that for every j lt 0



Set~t = sup

x 2 (max

iff i(x SE(A x) frac14 M (x))g)


brvbar ( frac14 j (x)) 6 ~t (415)

for almost all points in The following corollary provides an upper bound on the limit superior of the L 1

norms of local stresses in the shy ne-scale limit

Corollary 46 Suppose it is known that ` = lim supj 1 k Agravecedil j

i brvbar ( frac14 cedil j )k 1 lt 1Suppose further that for an interval 0 lt macr lt b the measure (volume) of the setswhere sup3 ìexcl macr i gt 0 does not vanish Then

lim supj 1

k Agravecedil j

i brvbar ( frac14 cedil j )k 1 6 kf i(x SE(A x) frac14 M (x))k 1 (416)

Corollary 46 follows upon setting t = ` iexcl macr and applying (314)For 1 6 p lt 1 we provide bounds on lim supj 1 k Agrave

cedil j

i brvbar ( frac14 cedil j )kp


942 R P Lipton

Corollary 47 Given that lim supj 1 k Agravecedil j

i brvbar ( frac14 deg j )k 1 lt 1

lim supj 1

k Agravecedil j

i brvbar ( frac14 cedil j )kp 6 kf i(x SE(A x) frac14 M (x))k 1 j laquo j1=p


This result follows from theorem 42 using the same arguments given to establishcorollary 36

(a) Proofs of theorems 41 and 42

To expedite the proofs we write the elements of the sequence fCmicro j


in terms of characteristic functions For the partition of laquo given byS

` = 1 Omicro j

` forwhich the diameters of O

micro j

` are less than microj we introduce the characteristic functionsAgrave

micro j

` such that Agravemicro j

` (x) = 1 for x in Omicro j

` and zero otherwise Then

Cmicro j

0 (A x x cedil j) =

NX

i

Agrave ji (x cedil jx)Ai (417)

where the characteristic function of the ith phase in the composite is specishy ed by

Agrave ji (x y) =

X

`

Agravemicro j

` (x) Agrave i(x`micro j

y) (418)

We start with the proof of theorem 42 Consider the set of points Scedil j

ti in theith phase for which brvbar ( frac14 cedil j ) gt t for a shy xed positive number t The characteristicfunction for S

cedil j

ti is denoted by Agrave jti(x) Passing to a subsequence if necessary one has

that Agrave jti

curren sup3 ti in L 1 ( laquo ) as j 1 Here the density sup3 ti(x) takes values in [0 1]

For any non-negative smooth test function p(x) one has

t

Z

laquo

pAgrave jti dx 6

Z

laquo

pAgrave jti brvbar ( frac14 cedil j ) dx (419)

Taking the limit on the left-hand side of (419) gives

t

Z

laquo

psup3 ti dx

As before the goal is to pass to the limit on the right-hand side of (419) to obtaina pointwise upper bound on the product tsup3 ti This will follow from the followinglemma


j = 1 and f frac14 cedil j g 1j = 1 also denoted

by fAgrave jtig1

j = 1 and f frac14 cedil j g 1j = 1 such that

Agrave jti brvbar ( frac14 cedil j ) AibrvbarAiHt(x)SE(A x) frac14 M (x) SE(A x)frac14 M (x)


AibrvbarAiHt(x)middotdeg middotdeg = limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(x r)


jrmiddotdeg )+middotdeg )) dy (420)



iexcl div(Cmicro j

0 (A y y cedil j)( deg (wjrmiddotdeg (y)) + middotdeg )) = 0 (421)

for y in Q(x r)



This lemma is established in the same way as lemma 37Passing to subsequences according to lemma 48 and on taking limits in (419) it

follows that


almost everywhere on laquo Passing to a diagonal sequence we have limj 1 rj = 0limj 1 microj=rj = 0 limj 1 (1=cedil j)microj = 0


micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj

middotdeg solves (421) with r = rj One can arrange things so that the cubesQ(x rj) contain the point x and an integral number of periods of side length 1=cedil j We change coordinates by translation and re-scaling so that Q(x rj) is transformedto the unit cube Q centred at the origin The change of variables is given byy = xj

0 + zrj where xj0 is the centre point of Q(x rj) and z lies in Q One can

write wjrj

middotdeg (y) = rj iquest j((y iexcl xj0)=rj) where iquest j(z) is the Q-periodic solution of

iexcl div(Cmicro j

0 (A xj0 + rjz rj cedil jz)( deg ( iquest j(z)) + middotdeg )) = 0 for z in Q (425)


ti(xj0 + rjz) and


Z

Q

Agrave jti(x

j0 + rjz) brvbar (Ai( deg ( iquest


Now introduce C0(A x rj cedil jz) =P

i Ai Agrave i(x rj cedil jz) and the Q-periodic solutions ~iquest j

ofiexcl div(C0(A x rj cedil jz)( deg ( ~iquest j(z)) + middotdeg )) = 0 for z in Q (427)

Recalling (42) it follows that

limj 1

Z

Q

jC micro j

0 (A xj0 + rjz rj cedil jz) iexcl C0(A x rj cedil jz)j dz = 0 (428)

Thus application of the higher integrability theorem of Meyers amp Elcrat (1975) (seeBensoussan et al 1978) shows that

limj 1

Z

Q

j deg ( iquest j(z)) iexcl deg (~iquest j(z))j2 dz = 0 (429)

Hence


Z

Q

Agrave jti(x

j0 + rjz) brvbar (Ai( deg (~iquest j(z)) + middotdeg )) dz (430)

One can write ~iquest j(z) = (1=(rj cedil j))rsquo(rj cedil jz) where rsquo(s) is the Q-periodic solution of

iexcl div(C0(A x s)( deg (rsquo(s)) + middotdeg )) = 0 for s in Q (431)


944 R P Lipton

It is noted that the sequence f deg ( ~iquest j(z))g 1j = 1 is equi-integrable since deg ( ~iquest j(z)) =

deg (rsquo(rj cedil jz)) Writing

Agrave ji (xj

0 + rjz rj cedil jz) = Agrave i(x rj cedil jz) + Agrave ji (xj

0 + rjz rj cedil jz) iexcl Agrave i(x rj cedil jz)

one notes from the equi-integrability of f deg ( ~iquest j(z))g 1j = 1 and (428) that


Z

Q

Agrave jti(x

j0 + rjz) Agrave i(x rj cedil jz) brvbar (Ai( deg ( ~iquest j(z)) + middotdeg )) dz

(432)Application of Holderrsquos inequality gives


6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz

parak Agrave i(x cent) brvbar (Ai( deg (rsquo(cent)) + middotdeg ))kL 1 (Q)

= sup3 ti(x)k Agrave i(x cent) brvbar (Ai( deg (rsquo(cent)) + middotdeg ))kL 1 (Q) (433)

Here we recall that both ~iquest j and rsquo are parametrized by x and theorem 42 follows from(422) The proof of theorem 42 for the case of piecewise constant locally periodicmicrostructures is simpler since iquest j = ~iquest j for sumacr ciently small cubes Q(x rj)

We give the proof of theorem 41 We consider a sequence of continuously gradedlocally periodic microstructures with elasticity tensors fC

micro j

0 (A x x cedil j)g 1j = 1 given by

(417) We prove that this sequence G-converges to CE(A x) given by (44)From the theory of G-convergence we can pass to a subsequence also denoted byfC

micro j

0 (A x x j)g 1j = 1 that G-converges to eCE(A x) Our shy rst goal is to show that

eCE(A x) = CE(A x)

To do this we use (210) to write

eCE(A x)middotdeg middotdeg

= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j

0 (A y y cedil j)( deg (wjrmiddotdeg ) + middotdeg ) ( deg (wjr


where wjrmiddotdeg solves (421) Passing to a diagonal sequence we have

limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j

paraAacutemicroj = 0

and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j

0 (A y y cedil j)( deg (wjrj

middotdeg ) + middotdeg ) ( deg (wjrj


Here wjrj

middotdeg solves (421) with r = rj As before one can arrange things so thatthe cubes Q(x rj) contain the point x and an integral number of periods of sidelength 1=cedil j We change coordinates by translation and rescaling so that Q(x rj) istransformed to the unit cube Q centred at the origin The change of variables is given



by y = xj0 + zrj where xj

0 is the centre point of Q(x rj) and z lies in Q One writesw

jrj

middotdeg (y) = rj iquest j((y iexcl xj0)=rj) where iquest j(z) is the solution of (425) This gives

eCE(A x)middotdeg middotdeg = limj 1

Z

Q

Cmicro j

0 (A xj0 + zrj rj cedil jz)(deg ( iquest j) + middotdeg ) ( deg ( iquest j) + middotdeg ) dz

We introduce C0(A x rj cedil jz) =P

i Ai Agrave i(x rj cedil jz) and the solutions ~iquest j of (427)Recalling (428) we see as before that

limj 1

Z

Q


Thus from the equi-integrability of the sequence f deg ( ~iquest j(z))g 1j = 1 and a change of vari-

ables we deduce that eCE(A x) = CE(A x) It is evident from the derivation givenabove that every subsequence of fC

micro j

0 (A x x cedil j)g1j = 1 has a subsequence that G-

converges to the same G-limit given by (44) Since the topology of G-convergence ismetrizable we deduce that the whole sequence fC

micro j

0 (A x x cedil j)g 1j = 1 G-converges to

(44) A standard application of the theorem of Meyers amp Elcrat (1975) shows thatCE(A x) is a continuous function of x Conversely given an enotective elastic tensorof the form (44) we can apply the procedure just used to construct a sequence ofcontinuously graded locally periodic microstructures that G-converge to it

This research ereg ort is sponsored by NSF through grant DMS-0072469 and by the Air Force Oplusmn ceof Scientimacrc Research Air Force Materiel Command USAF under grant number F49620-02-1-0041 The US Government is authorized to reproduce and distribute reprints for governmentalpurposes notwithstanding any copyright notation thereon The views and conclusions herein arethose of the authors and should not be interpreted as necessarily representing the oplusmn cial policiesor endorsements either expressed or implied of the Air Force Oplusmn ce of Scientimacrc Research orthe US Government

References

Bensoussan A Lions J L amp Papanicolaou G 1978 Asymptotic analysis for periodic structuresStudies in Mathematics and its Applications vol 5 Amsterdam North-Holland

Biot M A 1966 Fundamental skin ereg ect in anisotropic solid mechanics Int J Solids Struct2 645663

Buryachenko V A 2001 Multiparticle ereg ective macreld and related methods in micromechanics ofcomposite materials Appl Mech Rev 54 147

Dal Maso G 1993 An introduction to iexcl -convegence Progress in Nonlinear Direg erential Equationsand Their Applications vol 8 Boston Birkhauser

Duvaut G amp Lions J L 1976 Inequalities in mechanics and physics Grundlehren der mathe-matischen Wissenschaften in Einzeldarstellungen vol 219 Springer

Evans L C 1990 Weak convergence methods for nonlinear partial direg erential equations CBMSRegional Conference Series in Mathematics vol 74 Providence RI American MathematicalSociety

Glasser A M 1997 The use of transient FGM interlayers for joining advanced ceramics Com-posites B 28 7184

Koizumi M 1997 FGM activities in Japan Composites B 28 14

Lipton R 2002 Relaxation through homogenization for optimal design problems with gradientconstraints J Optim Theory Applicat 114 2753

Lipton R 2003 Stress constrained G closure and relaxation of structural design problems QAppl Math (In the press)


946 R P Lipton

Meyers N G amp Elcrat A 1975 Some results on regularity of non-linear elliptic equations andquasi-regular functions Duke Math J 47 121136

Milton G W 2002 The theory of composites Applied and Computational Mathematics vol 6Cambridge University Press

Murat F amp Tartar L 1997 H convergence In Topics in the mathematical modelling of compositematerials (ed A V Cherkaev amp R V Kohn) pp 2143 Boston Birkhauser

Pagano N amp Rybicky E F 1974 On the signimacrcance of the ereg ective modulus solutions formacrbrous composites J Composite Mater 8 214228

Pagano N amp Yuan F G 2000 The signimacrcance of the ereg ective modulus theory (homogenization)in composite laminate mechanics Composites Sci Technol 60 24712488

Ponte Casta~neda P amp Suquet P 1998 Nonlinear composites Adv Appl Mech 34 171302

Simon L 1979 On G-convergence of elliptic operators Indiana Univ Math J 28 587594

Spagnolo S 1976 Convergence in energy for elliptic operators In Proc 3rd Symp on NumericalSolutions of Partial Direg erential Equations (ed B Hubbard) pp 469498 Academic

Tartar L 2000 An introduction to the homogenization method in optimal design Springer Lec-ture Notes in Mathematics vol 1740 pp 45156

Tsai S W amp Hahn H T 1980 Introduction to composite materials Lancaster PA TechnomicPublishing Company


922 R P Lipton

suitable macroscopic quantities that can be used to assess the state of stress withina composite

In this treatment no assumption on the microstructure is made and the homoge-nized formula for the mean-square stress reguctuation is obtained (see theorem 23) Itis shown that the formula provides a lower bound on the limit inferior of the sequenceof L 1 norms of stresses associated with the discrete microstructure in the shy ne-scalelimit (see theorem 24) Mean-square stress reguctuations appear in many contextsincluding bounds for nonlinear composites (Ponte Casta~neda amp Suquet 1998) andestimates for failure surfaces for statistically homogeneous random composites (Bury-achenko 2001)

To obtain upper bounds on Lp norms 1 6 p 6 1 new quantities for the assessmentof the actual stress are introduced see (15) (34)(37) and (49) These quantitiesare referred to as macrostress modulation functions They measure the amplishy ca-tion or diminution of the macrostress by the underlying microgeometry For simplehypotheses on the asymptotic distribution of states for the stress these quantitiesare used to obtain upper bounds on the limit superior of the sequence of L1 norms ofstresses associated with the discrete microstructure in the shy ne-scale limit (see corol-laries 35 and 46) For 1 6 p lt 1 corollaries 36 and 47 provide upper bounds onthe limit superior of the sequence of Lp norms of stresses associated with the discretemicrostructure in the shy ne-scale limit For sumacr ciently small pointwise bounds onthe actual stress are given in terms of the macrostress modulation function (see (115)and corollaries 34 and 45) These results follow from a homogenization constraintthat correlates the macrostress modulation functions with the homogenized distribu-tion of states for the stress (see theorems 31 and 42) We apply the homogenizationconstraint to assess the size of sets over which the local stress in the compositeexceeds a prescribed value (see corollaries 32 33 43 and 44) The principal toolused to derive the constraint is a local formula for the derivative of enotective prop-erties given in theorem 22 The methods developed here can be applied to obtainanalogous results for local regux and gradient shy elds in the contexts of heat conductionDC electric transport and problems of dinotusion

Composites made from N linearly elastic materials are considered The domaincontaining the composite is denoted by laquo and the stress tensor inside the compositeat the point x is denoted by frac14 (x) Failure criteria are often given in terms of a co-ordinate invariant measure of the stress In this treatment we consider the equivalentstress given by

brvbar ( frac14 ) = brvbar frac14 frac14 (11)

where brvbar is a positive deshy nite fourth rank tensor (see Tsai amp Hahn 1980) Examplesof (11) include the Von Mises equivalent stress given by

brvbar ( frac14 ) = 12[( frac14



11 iexcl frac14 33)2]+3(( frac14

12)2+(frac14 13)2+(frac14

23)2) (12)

and the magnitude of the stress given by brvbar ( frac14 ) = j frac14 j2 =P3

ij = 1( frac14 ij)2 The set in

the ith phase where brvbar ( frac14 ) gt t is denoted by Sti The stress distribution function

para i (t) is the volume (measure) of S

ti One of the principal results of this paper istheorem 31 It provides a constraint relating macroscopic stress modulation functionand S

ti in the limit of vanishing To motivate theorem 31 and the macrostress modulation function we consider a

composite with uniform periodic microstructure contained in the unit cube Q The


















para k

i (t) = para 1i (t) = sup3 ti (18)



924 R P Lipton







is that the sets Sk



frac14 M (x) = CE deg (u M (x)) (111)





i (t) =R

laquoAgrave k


i (t) =R















~t = supx 2

(maxi



brvbar ( frac14 j (x)) 6 ~t (116)











lim supk 1

k Agrave k




926 R P Lipton





lim supk 1

k Agrave k






Agrave k


ti gt 0 (120)




in Sk








is identishy ed as











PNi= 1 Agrave





frac14 (x) = C(A x) deg (u(x)) (22)


deg (u) = 12(u

ij + uji) (23)


928 R P Lipton













C(P x) =NX

i= 1


NX

i = 1

Agrave i Ai



middotdeg where wr








CE(A x)


=3X

k = 1

3X

l = 1

3X

m= 1

3X

n= 1

[Mi]klmn limr 0

lim 0

micro1

jQ(x r)j

para

poundZ

Q(x r)

Agrave i ( deg (wr





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr


and

riklklC


= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr




lim 0

micro1

jQ(x r)j

para Z

Q(x r)











930 R P Lipton



lim 0

Z

Q(x r)






curren sup3 i





NX

i= 1


degdegdegdeg1

6 lim inf 0



13frac14

23 One has2X

i = 1

brvbarHi =G1 + G2

2p

G1G2

(215)

andG1 + G2

2p

G1G2

k( frac14 M13)2 + ( frac14 M

23)2k 1 6 lim inf 0

k( frac14 13)2 + ( frac14

23)2k 1 (216)







Z

Q(xr)



Z

Q(xr)




and Z

Q(xr)



jMij =

s X

klmn

[Mi]2klmn






Q(xr)





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



iexcl limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)



(221)

Writing





0 =

Z

Q(x r)



+

Z

Q(x r)




lim 0

micro1

jQ(x r)j

para Z

Q(x r)



+ macr shy limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



(223)


932 R P Lipton


Q(xr)



1

jQ(x r)j

para Z

Q(xr)






lim 0

micro1

jQ(x r)j

para Z

Q(x r)






Z

laquo




laquo

p(x)sup3



acutedx 6 0 (228)





Cj (A x) =

NX

i= 1

Agravej

i Ai

where Agravej













Consider the set Sj


limj 1

jSj

ti j =

Z

sup3 ti dx (33)



Cj (A x + rjz) =

NX

i = 1

Agravej

i (x + rjz)Ai




j


= Agravej





Agravej





limj 1

rj = 0 and limj 1

j=rj = 0


934 R P Lipton

Set


= supfrj g1

j=12 C

nlim sup

j 1kF i


o (37)





then we have







j








j











Set~t = sup

x 2

sup3max


acute


brvbar ( frac14 j (x)) 6 ~t (313)




fCjk (A x )g 1k = 1




in A =T 1

K = 1

S 1kgt K S


k gt K But

jAj 6macrmacr

1[

kgt ~K

Sjkt

macrmacr 6

1X

k = ~K










lim supj 1

kAgravedeg j



deg j




lim supj 1

k Agravedeg j








936 R P Lipton




laquo

Agravej


Z 1

0




lim supj 1

Z

laquo

Agravej


Z 1

0

tpiexcl1 lim supj 1

( para ji (t)) dt (317)


lim supj 1

para ji (t) =

Z

laquo

sup3 ti(x) dx


lim supj 1

Z

laquo

Agravej


Z ~J

0

tpiexcl1

Z

laquo







j





t

Z

laquo

pAgrave jti dx 6

Z

laquo







by fAgrave jtig1





= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)




= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)









j


j


Cj (P PN + 1 x) =

NX

m= 1m 6= i


j


ti(x)PN + 1


fCj (P PN + 1 x)g1j = 1






micro1

jQ(x rj)j

para Z

Q(x rj )


jrj



limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj


jrj




ti(x + rjz) and


Z

Q




938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton




























para k

i (t) = para 1i (t) = sup3 ti (18)



924 R P Lipton







is that the sets Sk



frac14 M (x) = CE deg (u M (x)) (111)





i (t) =R

laquoAgrave k


i (t) =R















~t = supx 2

(maxi



brvbar ( frac14 j (x)) 6 ~t (116)











lim supk 1

k Agrave k




926 R P Lipton





lim supk 1

k Agrave k






Agrave k


ti gt 0 (120)




in Sk








is identishy ed as











PNi= 1 Agrave





frac14 (x) = C(A x) deg (u(x)) (22)


deg (u) = 12(u

ij + uji) (23)


928 R P Lipton













C(P x) =NX

i= 1


NX

i = 1

Agrave i Ai



middotdeg where wr








CE(A x)


=3X

k = 1

3X

l = 1

3X

m= 1

3X

n= 1

[Mi]klmn limr 0

lim 0

micro1

jQ(x r)j

para

poundZ

Q(x r)

Agrave i ( deg (wr





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr


and

riklklC


= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr




lim 0

micro1

jQ(x r)j

para Z

Q(x r)











930 R P Lipton



lim 0

Z

Q(x r)






curren sup3 i





NX

i= 1


degdegdegdeg1

6 lim inf 0



13frac14

23 One has2X

i = 1

brvbarHi =G1 + G2

2p

G1G2

(215)

andG1 + G2

2p

G1G2

k( frac14 M13)2 + ( frac14 M

23)2k 1 6 lim inf 0

k( frac14 13)2 + ( frac14

23)2k 1 (216)







Z

Q(xr)



Z

Q(xr)




and Z

Q(xr)



jMij =

s X

klmn

[Mi]2klmn






Q(xr)





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



iexcl limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)



(221)

Writing





0 =

Z

Q(x r)



+

Z

Q(x r)




lim 0

micro1

jQ(x r)j

para Z

Q(x r)



+ macr shy limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



(223)


932 R P Lipton


Q(xr)



1

jQ(x r)j

para Z

Q(xr)






lim 0

micro1

jQ(x r)j

para Z

Q(x r)






Z

laquo




laquo

p(x)sup3



acutedx 6 0 (228)





Cj (A x) =

NX

i= 1

Agravej

i Ai

where Agravej













Consider the set Sj


limj 1

jSj

ti j =

Z

sup3 ti dx (33)



Cj (A x + rjz) =

NX

i = 1

Agravej

i (x + rjz)Ai




j


= Agravej





Agravej





limj 1

rj = 0 and limj 1

j=rj = 0


934 R P Lipton

Set


= supfrj g1

j=12 C

nlim sup

j 1kF i


o (37)





then we have







j








j











Set~t = sup

x 2

sup3max


acute


brvbar ( frac14 j (x)) 6 ~t (313)




fCjk (A x )g 1k = 1




in A =T 1

K = 1

S 1kgt K S


k gt K But

jAj 6macrmacr

1[

kgt ~K

Sjkt

macrmacr 6

1X

k = ~K










lim supj 1

kAgravedeg j



deg j




lim supj 1

k Agravedeg j








936 R P Lipton




laquo

Agravej


Z 1

0




lim supj 1

Z

laquo

Agravej


Z 1

0

tpiexcl1 lim supj 1

( para ji (t)) dt (317)


lim supj 1

para ji (t) =

Z

laquo

sup3 ti(x) dx


lim supj 1

Z

laquo

Agravej


Z ~J

0

tpiexcl1

Z

laquo







j





t

Z

laquo

pAgrave jti dx 6

Z

laquo







by fAgrave jtig1





= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)




= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)









j


j


Cj (P PN + 1 x) =

NX

m= 1m 6= i


j


ti(x)PN + 1


fCj (P PN + 1 x)g1j = 1






micro1

jQ(x rj)j

para Z

Q(x rj )


jrj



limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj


jrj




ti(x + rjz) and


Z

Q




938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton












924 R P Lipton







is that the sets Sk



frac14 M (x) = CE deg (u M (x)) (111)





i (t) =R

laquoAgrave k


i (t) =R















~t = supx 2

(maxi



brvbar ( frac14 j (x)) 6 ~t (116)











lim supk 1

k Agrave k




926 R P Lipton





lim supk 1

k Agrave k






Agrave k


ti gt 0 (120)




in Sk








is identishy ed as











PNi= 1 Agrave





frac14 (x) = C(A x) deg (u(x)) (22)


deg (u) = 12(u

ij + uji) (23)


928 R P Lipton













C(P x) =NX

i= 1


NX

i = 1

Agrave i Ai



middotdeg where wr








CE(A x)


=3X

k = 1

3X

l = 1

3X

m= 1

3X

n= 1

[Mi]klmn limr 0

lim 0

micro1

jQ(x r)j

para

poundZ

Q(x r)

Agrave i ( deg (wr





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr


and

riklklC


= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr




lim 0

micro1

jQ(x r)j

para Z

Q(x r)











930 R P Lipton



lim 0

Z

Q(x r)






curren sup3 i





NX

i= 1


degdegdegdeg1

6 lim inf 0



13frac14

23 One has2X

i = 1

brvbarHi =G1 + G2

2p

G1G2

(215)

andG1 + G2

2p

G1G2

k( frac14 M13)2 + ( frac14 M

23)2k 1 6 lim inf 0

k( frac14 13)2 + ( frac14

23)2k 1 (216)







Z

Q(xr)



Z

Q(xr)




and Z

Q(xr)



jMij =

s X

klmn

[Mi]2klmn






Q(xr)





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



iexcl limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)



(221)

Writing





0 =

Z

Q(x r)



+

Z

Q(x r)




lim 0

micro1

jQ(x r)j

para Z

Q(x r)



+ macr shy limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



(223)


932 R P Lipton


Q(xr)



1

jQ(x r)j

para Z

Q(xr)






lim 0

micro1

jQ(x r)j

para Z

Q(x r)






Z

laquo




laquo

p(x)sup3



acutedx 6 0 (228)





Cj (A x) =

NX

i= 1

Agravej

i Ai

where Agravej













Consider the set Sj


limj 1

jSj

ti j =

Z

sup3 ti dx (33)



Cj (A x + rjz) =

NX

i = 1

Agravej

i (x + rjz)Ai




j


= Agravej





Agravej





limj 1

rj = 0 and limj 1

j=rj = 0


934 R P Lipton

Set


= supfrj g1

j=12 C

nlim sup

j 1kF i


o (37)





then we have







j








j











Set~t = sup

x 2

sup3max


acute


brvbar ( frac14 j (x)) 6 ~t (313)




fCjk (A x )g 1k = 1




in A =T 1

K = 1

S 1kgt K S


k gt K But

jAj 6macrmacr

1[

kgt ~K

Sjkt

macrmacr 6

1X

k = ~K










lim supj 1

kAgravedeg j



deg j




lim supj 1

k Agravedeg j








936 R P Lipton




laquo

Agravej


Z 1

0




lim supj 1

Z

laquo

Agravej


Z 1

0

tpiexcl1 lim supj 1

( para ji (t)) dt (317)


lim supj 1

para ji (t) =

Z

laquo

sup3 ti(x) dx


lim supj 1

Z

laquo

Agravej


Z ~J

0

tpiexcl1

Z

laquo







j





t

Z

laquo

pAgrave jti dx 6

Z

laquo







by fAgrave jtig1





= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)




= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)









j


j


Cj (P PN + 1 x) =

NX

m= 1m 6= i


j


ti(x)PN + 1


fCj (P PN + 1 x)g1j = 1






micro1

jQ(x rj)j

para Z

Q(x rj )


jrj



limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj


jrj




ti(x + rjz) and


Z

Q




938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton




















~t = supx 2

(maxi



brvbar ( frac14 j (x)) 6 ~t (116)











lim supk 1

k Agrave k




926 R P Lipton





lim supk 1

k Agrave k






Agrave k


ti gt 0 (120)




in Sk








is identishy ed as











PNi= 1 Agrave





frac14 (x) = C(A x) deg (u(x)) (22)


deg (u) = 12(u

ij + uji) (23)


928 R P Lipton













C(P x) =NX

i= 1


NX

i = 1

Agrave i Ai



middotdeg where wr








CE(A x)


=3X

k = 1

3X

l = 1

3X

m= 1

3X

n= 1

[Mi]klmn limr 0

lim 0

micro1

jQ(x r)j

para

poundZ

Q(x r)

Agrave i ( deg (wr





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr


and

riklklC


= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr




lim 0

micro1

jQ(x r)j

para Z

Q(x r)











930 R P Lipton



lim 0

Z

Q(x r)






curren sup3 i





NX

i= 1


degdegdegdeg1

6 lim inf 0



13frac14

23 One has2X

i = 1

brvbarHi =G1 + G2

2p

G1G2

(215)

andG1 + G2

2p

G1G2

k( frac14 M13)2 + ( frac14 M

23)2k 1 6 lim inf 0

k( frac14 13)2 + ( frac14

23)2k 1 (216)







Z

Q(xr)



Z

Q(xr)




and Z

Q(xr)



jMij =

s X

klmn

[Mi]2klmn






Q(xr)





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



iexcl limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)



(221)

Writing





0 =

Z

Q(x r)



+

Z

Q(x r)




lim 0

micro1

jQ(x r)j

para Z

Q(x r)



+ macr shy limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



(223)


932 R P Lipton


Q(xr)



1

jQ(x r)j

para Z

Q(xr)






lim 0

micro1

jQ(x r)j

para Z

Q(x r)






Z

laquo




laquo

p(x)sup3



acutedx 6 0 (228)





Cj (A x) =

NX

i= 1

Agravej

i Ai

where Agravej













Consider the set Sj


limj 1

jSj

ti j =

Z

sup3 ti dx (33)



Cj (A x + rjz) =

NX

i = 1

Agravej

i (x + rjz)Ai




j


= Agravej





Agravej





limj 1

rj = 0 and limj 1

j=rj = 0


934 R P Lipton

Set


= supfrj g1

j=12 C

nlim sup

j 1kF i


o (37)





then we have







j








j











Set~t = sup

x 2

sup3max


acute


brvbar ( frac14 j (x)) 6 ~t (313)




fCjk (A x )g 1k = 1




in A =T 1

K = 1

S 1kgt K S


k gt K But

jAj 6macrmacr

1[

kgt ~K

Sjkt

macrmacr 6

1X

k = ~K










lim supj 1

kAgravedeg j



deg j




lim supj 1

k Agravedeg j








936 R P Lipton




laquo

Agravej


Z 1

0




lim supj 1

Z

laquo

Agravej


Z 1

0

tpiexcl1 lim supj 1

( para ji (t)) dt (317)


lim supj 1

para ji (t) =

Z

laquo

sup3 ti(x) dx


lim supj 1

Z

laquo

Agravej


Z ~J

0

tpiexcl1

Z

laquo







j





t

Z

laquo

pAgrave jti dx 6

Z

laquo







by fAgrave jtig1





= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)




= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)









j


j


Cj (P PN + 1 x) =

NX

m= 1m 6= i


j


ti(x)PN + 1


fCj (P PN + 1 x)g1j = 1






micro1

jQ(x rj)j

para Z

Q(x rj )


jrj



limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj


jrj




ti(x + rjz) and


Z

Q




938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton












926 R P Lipton





lim supk 1

k Agrave k






Agrave k


ti gt 0 (120)




in Sk








is identishy ed as











PNi= 1 Agrave





frac14 (x) = C(A x) deg (u(x)) (22)


deg (u) = 12(u

ij + uji) (23)


928 R P Lipton













C(P x) =NX

i= 1


NX

i = 1

Agrave i Ai



middotdeg where wr








CE(A x)


=3X

k = 1

3X

l = 1

3X

m= 1

3X

n= 1

[Mi]klmn limr 0

lim 0

micro1

jQ(x r)j

para

poundZ

Q(x r)

Agrave i ( deg (wr





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr


and

riklklC


= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr




lim 0

micro1

jQ(x r)j

para Z

Q(x r)











930 R P Lipton



lim 0

Z

Q(x r)






curren sup3 i





NX

i= 1


degdegdegdeg1

6 lim inf 0



13frac14

23 One has2X

i = 1

brvbarHi =G1 + G2

2p

G1G2

(215)

andG1 + G2

2p

G1G2

k( frac14 M13)2 + ( frac14 M

23)2k 1 6 lim inf 0

k( frac14 13)2 + ( frac14

23)2k 1 (216)







Z

Q(xr)



Z

Q(xr)




and Z

Q(xr)



jMij =

s X

klmn

[Mi]2klmn






Q(xr)





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



iexcl limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)



(221)

Writing





0 =

Z

Q(x r)



+

Z

Q(x r)




lim 0

micro1

jQ(x r)j

para Z

Q(x r)



+ macr shy limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



(223)


932 R P Lipton


Q(xr)



1

jQ(x r)j

para Z

Q(xr)






lim 0

micro1

jQ(x r)j

para Z

Q(x r)






Z

laquo




laquo

p(x)sup3



acutedx 6 0 (228)





Cj (A x) =

NX

i= 1

Agravej

i Ai

where Agravej













Consider the set Sj


limj 1

jSj

ti j =

Z

sup3 ti dx (33)



Cj (A x + rjz) =

NX

i = 1

Agravej

i (x + rjz)Ai




j


= Agravej





Agravej





limj 1

rj = 0 and limj 1

j=rj = 0


934 R P Lipton

Set


= supfrj g1

j=12 C

nlim sup

j 1kF i


o (37)





then we have







j








j











Set~t = sup

x 2

sup3max


acute


brvbar ( frac14 j (x)) 6 ~t (313)




fCjk (A x )g 1k = 1




in A =T 1

K = 1

S 1kgt K S


k gt K But

jAj 6macrmacr

1[

kgt ~K

Sjkt

macrmacr 6

1X

k = ~K










lim supj 1

kAgravedeg j



deg j




lim supj 1

k Agravedeg j








936 R P Lipton




laquo

Agravej


Z 1

0




lim supj 1

Z

laquo

Agravej


Z 1

0

tpiexcl1 lim supj 1

( para ji (t)) dt (317)


lim supj 1

para ji (t) =

Z

laquo

sup3 ti(x) dx


lim supj 1

Z

laquo

Agravej


Z ~J

0

tpiexcl1

Z

laquo







j





t

Z

laquo

pAgrave jti dx 6

Z

laquo







by fAgrave jtig1





= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)




= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)









j


j


Cj (P PN + 1 x) =

NX

m= 1m 6= i


j


ti(x)PN + 1


fCj (P PN + 1 x)g1j = 1






micro1

jQ(x rj)j

para Z

Q(x rj )


jrj



limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj


jrj




ti(x + rjz) and


Z

Q




938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton
















is identishy ed as











PNi= 1 Agrave





frac14 (x) = C(A x) deg (u(x)) (22)


deg (u) = 12(u

ij + uji) (23)


928 R P Lipton













C(P x) =NX

i= 1


NX

i = 1

Agrave i Ai



middotdeg where wr








CE(A x)


=3X

k = 1

3X

l = 1

3X

m= 1

3X

n= 1

[Mi]klmn limr 0

lim 0

micro1

jQ(x r)j

para

poundZ

Q(x r)

Agrave i ( deg (wr





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr


and

riklklC


= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr




lim 0

micro1

jQ(x r)j

para Z

Q(x r)











930 R P Lipton



lim 0

Z

Q(x r)






curren sup3 i





NX

i= 1


degdegdegdeg1

6 lim inf 0



13frac14

23 One has2X

i = 1

brvbarHi =G1 + G2

2p

G1G2

(215)

andG1 + G2

2p

G1G2

k( frac14 M13)2 + ( frac14 M

23)2k 1 6 lim inf 0

k( frac14 13)2 + ( frac14

23)2k 1 (216)







Z

Q(xr)



Z

Q(xr)




and Z

Q(xr)



jMij =

s X

klmn

[Mi]2klmn






Q(xr)





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



iexcl limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)



(221)

Writing





0 =

Z

Q(x r)



+

Z

Q(x r)




lim 0

micro1

jQ(x r)j

para Z

Q(x r)



+ macr shy limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



(223)


932 R P Lipton


Q(xr)



1

jQ(x r)j

para Z

Q(xr)






lim 0

micro1

jQ(x r)j

para Z

Q(x r)






Z

laquo




laquo

p(x)sup3



acutedx 6 0 (228)





Cj (A x) =

NX

i= 1

Agravej

i Ai

where Agravej













Consider the set Sj


limj 1

jSj

ti j =

Z

sup3 ti dx (33)



Cj (A x + rjz) =

NX

i = 1

Agravej

i (x + rjz)Ai




j


= Agravej





Agravej





limj 1

rj = 0 and limj 1

j=rj = 0


934 R P Lipton

Set


= supfrj g1

j=12 C

nlim sup

j 1kF i


o (37)





then we have







j








j











Set~t = sup

x 2

sup3max


acute


brvbar ( frac14 j (x)) 6 ~t (313)




fCjk (A x )g 1k = 1




in A =T 1

K = 1

S 1kgt K S


k gt K But

jAj 6macrmacr

1[

kgt ~K

Sjkt

macrmacr 6

1X

k = ~K










lim supj 1

kAgravedeg j



deg j




lim supj 1

k Agravedeg j








936 R P Lipton




laquo

Agravej


Z 1

0




lim supj 1

Z

laquo

Agravej


Z 1

0

tpiexcl1 lim supj 1

( para ji (t)) dt (317)


lim supj 1

para ji (t) =

Z

laquo

sup3 ti(x) dx


lim supj 1

Z

laquo

Agravej


Z ~J

0

tpiexcl1

Z

laquo







j





t

Z

laquo

pAgrave jti dx 6

Z

laquo







by fAgrave jtig1





= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)




= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)









j


j


Cj (P PN + 1 x) =

NX

m= 1m 6= i


j


ti(x)PN + 1


fCj (P PN + 1 x)g1j = 1






micro1

jQ(x rj)j

para Z

Q(x rj )


jrj



limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj


jrj




ti(x + rjz) and


Z

Q




938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton












928 R P Lipton













C(P x) =NX

i= 1


NX

i = 1

Agrave i Ai



middotdeg where wr








CE(A x)


=3X

k = 1

3X

l = 1

3X

m= 1

3X

n= 1

[Mi]klmn limr 0

lim 0

micro1

jQ(x r)j

para

poundZ

Q(x r)

Agrave i ( deg (wr





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr


and

riklklC


= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr




lim 0

micro1

jQ(x r)j

para Z

Q(x r)











930 R P Lipton



lim 0

Z

Q(x r)






curren sup3 i





NX

i= 1


degdegdegdeg1

6 lim inf 0



13frac14

23 One has2X

i = 1

brvbarHi =G1 + G2

2p

G1G2

(215)

andG1 + G2

2p

G1G2

k( frac14 M13)2 + ( frac14 M

23)2k 1 6 lim inf 0

k( frac14 13)2 + ( frac14

23)2k 1 (216)







Z

Q(xr)



Z

Q(xr)




and Z

Q(xr)



jMij =

s X

klmn

[Mi]2klmn






Q(xr)





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



iexcl limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)



(221)

Writing





0 =

Z

Q(x r)



+

Z

Q(x r)




lim 0

micro1

jQ(x r)j

para Z

Q(x r)



+ macr shy limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



(223)


932 R P Lipton


Q(xr)



1

jQ(x r)j

para Z

Q(xr)






lim 0

micro1

jQ(x r)j

para Z

Q(x r)






Z

laquo




laquo

p(x)sup3



acutedx 6 0 (228)





Cj (A x) =

NX

i= 1

Agravej

i Ai

where Agravej













Consider the set Sj


limj 1

jSj

ti j =

Z

sup3 ti dx (33)



Cj (A x + rjz) =

NX

i = 1

Agravej

i (x + rjz)Ai




j


= Agravej





Agravej





limj 1

rj = 0 and limj 1

j=rj = 0


934 R P Lipton

Set


= supfrj g1

j=12 C

nlim sup

j 1kF i


o (37)





then we have







j








j











Set~t = sup

x 2

sup3max


acute


brvbar ( frac14 j (x)) 6 ~t (313)




fCjk (A x )g 1k = 1




in A =T 1

K = 1

S 1kgt K S


k gt K But

jAj 6macrmacr

1[

kgt ~K

Sjkt

macrmacr 6

1X

k = ~K










lim supj 1

kAgravedeg j



deg j




lim supj 1

k Agravedeg j








936 R P Lipton




laquo

Agravej


Z 1

0




lim supj 1

Z

laquo

Agravej


Z 1

0

tpiexcl1 lim supj 1

( para ji (t)) dt (317)


lim supj 1

para ji (t) =

Z

laquo

sup3 ti(x) dx


lim supj 1

Z

laquo

Agravej


Z ~J

0

tpiexcl1

Z

laquo







j





t

Z

laquo

pAgrave jti dx 6

Z

laquo







by fAgrave jtig1





= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)




= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)









j


j


Cj (P PN + 1 x) =

NX

m= 1m 6= i


j


ti(x)PN + 1


fCj (P PN + 1 x)g1j = 1






micro1

jQ(x rj)j

para Z

Q(x rj )


jrj



limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj


jrj




ti(x + rjz) and


Z

Q




938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton














CE(A x)


=3X

k = 1

3X

l = 1

3X

m= 1

3X

n= 1

[Mi]klmn limr 0

lim 0

micro1

jQ(x r)j

para

poundZ

Q(x r)

Agrave i ( deg (wr





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr


and

riklklC


= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)

Agrave i ( deg (wr




lim 0

micro1

jQ(x r)j

para Z

Q(x r)











930 R P Lipton



lim 0

Z

Q(x r)






curren sup3 i





NX

i= 1


degdegdegdeg1

6 lim inf 0



13frac14

23 One has2X

i = 1

brvbarHi =G1 + G2

2p

G1G2

(215)

andG1 + G2

2p

G1G2

k( frac14 M13)2 + ( frac14 M

23)2k 1 6 lim inf 0

k( frac14 13)2 + ( frac14

23)2k 1 (216)







Z

Q(xr)



Z

Q(xr)




and Z

Q(xr)



jMij =

s X

klmn

[Mi]2klmn






Q(xr)





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



iexcl limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)



(221)

Writing





0 =

Z

Q(x r)



+

Z

Q(x r)




lim 0

micro1

jQ(x r)j

para Z

Q(x r)



+ macr shy limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



(223)


932 R P Lipton


Q(xr)



1

jQ(x r)j

para Z

Q(xr)






lim 0

micro1

jQ(x r)j

para Z

Q(x r)






Z

laquo




laquo

p(x)sup3



acutedx 6 0 (228)





Cj (A x) =

NX

i= 1

Agravej

i Ai

where Agravej













Consider the set Sj


limj 1

jSj

ti j =

Z

sup3 ti dx (33)



Cj (A x + rjz) =

NX

i = 1

Agravej

i (x + rjz)Ai




j


= Agravej





Agravej





limj 1

rj = 0 and limj 1

j=rj = 0


934 R P Lipton

Set


= supfrj g1

j=12 C

nlim sup

j 1kF i


o (37)





then we have







j








j











Set~t = sup

x 2

sup3max


acute


brvbar ( frac14 j (x)) 6 ~t (313)




fCjk (A x )g 1k = 1




in A =T 1

K = 1

S 1kgt K S


k gt K But

jAj 6macrmacr

1[

kgt ~K

Sjkt

macrmacr 6

1X

k = ~K










lim supj 1

kAgravedeg j



deg j




lim supj 1

k Agravedeg j








936 R P Lipton




laquo

Agravej


Z 1

0




lim supj 1

Z

laquo

Agravej


Z 1

0

tpiexcl1 lim supj 1

( para ji (t)) dt (317)


lim supj 1

para ji (t) =

Z

laquo

sup3 ti(x) dx


lim supj 1

Z

laquo

Agravej


Z ~J

0

tpiexcl1

Z

laquo







j





t

Z

laquo

pAgrave jti dx 6

Z

laquo







by fAgrave jtig1





= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)




= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)









j


j


Cj (P PN + 1 x) =

NX

m= 1m 6= i


j


ti(x)PN + 1


fCj (P PN + 1 x)g1j = 1






micro1

jQ(x rj)j

para Z

Q(x rj )


jrj



limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj


jrj




ti(x + rjz) and


Z

Q




938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton












930 R P Lipton



lim 0

Z

Q(x r)






curren sup3 i





NX

i= 1


degdegdegdeg1

6 lim inf 0



13frac14

23 One has2X

i = 1

brvbarHi =G1 + G2

2p

G1G2

(215)

andG1 + G2

2p

G1G2

k( frac14 M13)2 + ( frac14 M

23)2k 1 6 lim inf 0

k( frac14 13)2 + ( frac14

23)2k 1 (216)







Z

Q(xr)



Z

Q(xr)




and Z

Q(xr)



jMij =

s X

klmn

[Mi]2klmn






Q(xr)





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



iexcl limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)



(221)

Writing





0 =

Z

Q(x r)



+

Z

Q(x r)




lim 0

micro1

jQ(x r)j

para Z

Q(x r)



+ macr shy limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



(223)


932 R P Lipton


Q(xr)



1

jQ(x r)j

para Z

Q(xr)






lim 0

micro1

jQ(x r)j

para Z

Q(x r)






Z

laquo




laquo

p(x)sup3



acutedx 6 0 (228)





Cj (A x) =

NX

i= 1

Agravej

i Ai

where Agravej













Consider the set Sj


limj 1

jSj

ti j =

Z

sup3 ti dx (33)



Cj (A x + rjz) =

NX

i = 1

Agravej

i (x + rjz)Ai




j


= Agravej





Agravej





limj 1

rj = 0 and limj 1

j=rj = 0


934 R P Lipton

Set


= supfrj g1

j=12 C

nlim sup

j 1kF i


o (37)





then we have







j








j











Set~t = sup

x 2

sup3max


acute


brvbar ( frac14 j (x)) 6 ~t (313)




fCjk (A x )g 1k = 1




in A =T 1

K = 1

S 1kgt K S


k gt K But

jAj 6macrmacr

1[

kgt ~K

Sjkt

macrmacr 6

1X

k = ~K










lim supj 1

kAgravedeg j



deg j




lim supj 1

k Agravedeg j








936 R P Lipton




laquo

Agravej


Z 1

0




lim supj 1

Z

laquo

Agravej


Z 1

0

tpiexcl1 lim supj 1

( para ji (t)) dt (317)


lim supj 1

para ji (t) =

Z

laquo

sup3 ti(x) dx


lim supj 1

Z

laquo

Agravej


Z ~J

0

tpiexcl1

Z

laquo







j





t

Z

laquo

pAgrave jti dx 6

Z

laquo







by fAgrave jtig1





= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)




= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)









j


j


Cj (P PN + 1 x) =

NX

m= 1m 6= i


j


ti(x)PN + 1


fCj (P PN + 1 x)g1j = 1






micro1

jQ(x rj)j

para Z

Q(x rj )


jrj



limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj


jrj




ti(x + rjz) and


Z

Q




938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton













and Z

Q(xr)



jMij =

s X

klmn

[Mi]2klmn






Q(xr)





= limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



iexcl limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(x r)



(221)

Writing





0 =

Z

Q(x r)



+

Z

Q(x r)




lim 0

micro1

jQ(x r)j

para Z

Q(x r)



+ macr shy limr 0

lim 0

micro1

jQ(x r)j

para Z

Q(xr)



(223)


932 R P Lipton


Q(xr)



1

jQ(x r)j

para Z

Q(xr)






lim 0

micro1

jQ(x r)j

para Z

Q(x r)






Z

laquo




laquo

p(x)sup3



acutedx 6 0 (228)





Cj (A x) =

NX

i= 1

Agravej

i Ai

where Agravej













Consider the set Sj


limj 1

jSj

ti j =

Z

sup3 ti dx (33)



Cj (A x + rjz) =

NX

i = 1

Agravej

i (x + rjz)Ai




j


= Agravej





Agravej





limj 1

rj = 0 and limj 1

j=rj = 0


934 R P Lipton

Set


= supfrj g1

j=12 C

nlim sup

j 1kF i


o (37)





then we have







j








j











Set~t = sup

x 2

sup3max


acute


brvbar ( frac14 j (x)) 6 ~t (313)




fCjk (A x )g 1k = 1




in A =T 1

K = 1

S 1kgt K S


k gt K But

jAj 6macrmacr

1[

kgt ~K

Sjkt

macrmacr 6

1X

k = ~K










lim supj 1

kAgravedeg j



deg j




lim supj 1

k Agravedeg j








936 R P Lipton




laquo

Agravej


Z 1

0




lim supj 1

Z

laquo

Agravej


Z 1

0

tpiexcl1 lim supj 1

( para ji (t)) dt (317)


lim supj 1

para ji (t) =

Z

laquo

sup3 ti(x) dx


lim supj 1

Z

laquo

Agravej


Z ~J

0

tpiexcl1

Z

laquo







j





t

Z

laquo

pAgrave jti dx 6

Z

laquo







by fAgrave jtig1





= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)




= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)









j


j


Cj (P PN + 1 x) =

NX

m= 1m 6= i


j


ti(x)PN + 1


fCj (P PN + 1 x)g1j = 1






micro1

jQ(x rj)j

para Z

Q(x rj )


jrj



limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj


jrj




ti(x + rjz) and


Z

Q




938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton












932 R P Lipton


Q(xr)



1

jQ(x r)j

para Z

Q(xr)






lim 0

micro1

jQ(x r)j

para Z

Q(x r)






Z

laquo




laquo

p(x)sup3



acutedx 6 0 (228)





Cj (A x) =

NX

i= 1

Agravej

i Ai

where Agravej













Consider the set Sj


limj 1

jSj

ti j =

Z

sup3 ti dx (33)



Cj (A x + rjz) =

NX

i = 1

Agravej

i (x + rjz)Ai




j


= Agravej





Agravej





limj 1

rj = 0 and limj 1

j=rj = 0


934 R P Lipton

Set


= supfrj g1

j=12 C

nlim sup

j 1kF i


o (37)





then we have







j








j











Set~t = sup

x 2

sup3max


acute


brvbar ( frac14 j (x)) 6 ~t (313)




fCjk (A x )g 1k = 1




in A =T 1

K = 1

S 1kgt K S


k gt K But

jAj 6macrmacr

1[

kgt ~K

Sjkt

macrmacr 6

1X

k = ~K










lim supj 1

kAgravedeg j



deg j




lim supj 1

k Agravedeg j








936 R P Lipton




laquo

Agravej


Z 1

0




lim supj 1

Z

laquo

Agravej


Z 1

0

tpiexcl1 lim supj 1

( para ji (t)) dt (317)


lim supj 1

para ji (t) =

Z

laquo

sup3 ti(x) dx


lim supj 1

Z

laquo

Agravej


Z ~J

0

tpiexcl1

Z

laquo







j





t

Z

laquo

pAgrave jti dx 6

Z

laquo







by fAgrave jtig1





= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)




= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)









j


j


Cj (P PN + 1 x) =

NX

m= 1m 6= i


j


ti(x)PN + 1


fCj (P PN + 1 x)g1j = 1






micro1

jQ(x rj)j

para Z

Q(x rj )


jrj



limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj


jrj




ti(x + rjz) and


Z

Q




938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton



















Consider the set Sj


limj 1

jSj

ti j =

Z

sup3 ti dx (33)



Cj (A x + rjz) =

NX

i = 1

Agravej

i (x + rjz)Ai




j


= Agravej





Agravej





limj 1

rj = 0 and limj 1

j=rj = 0


934 R P Lipton

Set


= supfrj g1

j=12 C

nlim sup

j 1kF i


o (37)





then we have







j








j











Set~t = sup

x 2

sup3max


acute


brvbar ( frac14 j (x)) 6 ~t (313)




fCjk (A x )g 1k = 1




in A =T 1

K = 1

S 1kgt K S


k gt K But

jAj 6macrmacr

1[

kgt ~K

Sjkt

macrmacr 6

1X

k = ~K










lim supj 1

kAgravedeg j



deg j




lim supj 1

k Agravedeg j








936 R P Lipton




laquo

Agravej


Z 1

0




lim supj 1

Z

laquo

Agravej


Z 1

0

tpiexcl1 lim supj 1

( para ji (t)) dt (317)


lim supj 1

para ji (t) =

Z

laquo

sup3 ti(x) dx


lim supj 1

Z

laquo

Agravej


Z ~J

0

tpiexcl1

Z

laquo







j





t

Z

laquo

pAgrave jti dx 6

Z

laquo







by fAgrave jtig1





= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)




= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)









j


j


Cj (P PN + 1 x) =

NX

m= 1m 6= i


j


ti(x)PN + 1


fCj (P PN + 1 x)g1j = 1






micro1

jQ(x rj)j

para Z

Q(x rj )


jrj



limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj


jrj




ti(x + rjz) and


Z

Q




938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton












934 R P Lipton

Set


= supfrj g1

j=12 C

nlim sup

j 1kF i


o (37)





then we have







j








j











Set~t = sup

x 2

sup3max


acute


brvbar ( frac14 j (x)) 6 ~t (313)




fCjk (A x )g 1k = 1




in A =T 1

K = 1

S 1kgt K S


k gt K But

jAj 6macrmacr

1[

kgt ~K

Sjkt

macrmacr 6

1X

k = ~K










lim supj 1

kAgravedeg j



deg j




lim supj 1

k Agravedeg j








936 R P Lipton




laquo

Agravej


Z 1

0




lim supj 1

Z

laquo

Agravej


Z 1

0

tpiexcl1 lim supj 1

( para ji (t)) dt (317)


lim supj 1

para ji (t) =

Z

laquo

sup3 ti(x) dx


lim supj 1

Z

laquo

Agravej


Z ~J

0

tpiexcl1

Z

laquo







j





t

Z

laquo

pAgrave jti dx 6

Z

laquo







by fAgrave jtig1





= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)




= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)









j


j


Cj (P PN + 1 x) =

NX

m= 1m 6= i


j


ti(x)PN + 1


fCj (P PN + 1 x)g1j = 1






micro1

jQ(x rj)j

para Z

Q(x rj )


jrj



limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj


jrj




ti(x + rjz) and


Z

Q




938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton













Set~t = sup

x 2

sup3max


acute


brvbar ( frac14 j (x)) 6 ~t (313)




fCjk (A x )g 1k = 1




in A =T 1

K = 1

S 1kgt K S


k gt K But

jAj 6macrmacr

1[

kgt ~K

Sjkt

macrmacr 6

1X

k = ~K










lim supj 1

kAgravedeg j



deg j




lim supj 1

k Agravedeg j








936 R P Lipton




laquo

Agravej


Z 1

0




lim supj 1

Z

laquo

Agravej


Z 1

0

tpiexcl1 lim supj 1

( para ji (t)) dt (317)


lim supj 1

para ji (t) =

Z

laquo

sup3 ti(x) dx


lim supj 1

Z

laquo

Agravej


Z ~J

0

tpiexcl1

Z

laquo







j





t

Z

laquo

pAgrave jti dx 6

Z

laquo







by fAgrave jtig1





= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)




= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)









j


j


Cj (P PN + 1 x) =

NX

m= 1m 6= i


j


ti(x)PN + 1


fCj (P PN + 1 x)g1j = 1






micro1

jQ(x rj)j

para Z

Q(x rj )


jrj



limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj


jrj




ti(x + rjz) and


Z

Q




938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton












936 R P Lipton




laquo

Agravej


Z 1

0




lim supj 1

Z

laquo

Agravej


Z 1

0

tpiexcl1 lim supj 1

( para ji (t)) dt (317)


lim supj 1

para ji (t) =

Z

laquo

sup3 ti(x) dx


lim supj 1

Z

laquo

Agravej


Z ~J

0

tpiexcl1

Z

laquo







j





t

Z

laquo

pAgrave jti dx 6

Z

laquo







by fAgrave jtig1





= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)




= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)









j


j


Cj (P PN + 1 x) =

NX

m= 1m 6= i


j


ti(x)PN + 1


fCj (P PN + 1 x)g1j = 1






micro1

jQ(x rj)j

para Z

Q(x rj )


jrj



limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj


jrj




ti(x + rjz) and


Z

Q




938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton

















j


j


Cj (P PN + 1 x) =

NX

m= 1m 6= i


j


ti(x)PN + 1


fCj (P PN + 1 x)g1j = 1






micro1

jQ(x rj)j

para Z

Q(x rj )


jrj



limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj


jrj




ti(x + rjz) and


Z

Q




938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton












938 R P Lipton



6micro

lim supj 1

micro1

jQj

para Z

Q


paralim sup

j 1kF i





j


j = 1 in C








C0(A x y) = C`0(A y) (41)










limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton














limh 0

Z

Q








micro j


micro j

0 (A x x j)g 1j = 1


C0(A x y) =

NX

i




Z

Q







micro j


following theorem




micro j




940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton












940 R P Lipton



micro j



iexcl div(Cmicro j



micro j


micro j
















micro j



then we have









cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton
















cedil j








cedil j









Set~t = sup

x 2 (max



brvbar ( frac14 j (x)) 6 ~t (415)





lim supj 1

k Agravecedil j



cedil j



942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton












942 R P Lipton



lim supj 1

k Agravecedil j








` = 1 Omicro j


micro j


micro j




Cmicro j

0 (A x x cedil j) =

NX

i



Agrave ji (x y) =

X

`

Agravemicro j


y) (418)



cedil j


that Agrave jti



t

Z

laquo

pAgrave jti dx 6

Z

laquo



t

Z

laquo

psup3 ti dx




by fAgrave jtig1





limj 1

micro1

jQ(x r)j

para Z

Q(x r)





iexcl div(Cmicro j


for y in Q(x r)




follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton














follows that




micro1

jQ(x rj)j

para Z

Q(x rj )


jrj


and

limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )


Here wjrj



write wjrj


iexcl div(Cmicro j



ti(xj0 + rjz) and


Z

Q

Agrave jti(x







limj 1

Z

Q

jC micro j



limj 1

Z

Q


Hence


Z

Q

Agrave jti(x





944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton












944 R P Lipton



Agrave ji (xj





Z

Q

Agrave jti(x




6micro

lim supj 1

micro1

jQj

para Z

Q

Agrave jti(x

j0 + rjz) dz





micro j



micro j


eCE(A x) = CE(A x)



= limr 0

limj 1

micro1

jQ(x r)j

para Z

Q(xr)

Cmicro j




limj 1

rj = 0 limj 1

microj

rj

= 0 limj 1

micro1

cedil j


and


= limj 1

micro1

jQ(x rj)j

para Z

Q(xrj )

Cmicro j




Here wjrj






jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton















jrj



Z

Q

Cmicro j




limj 1

Z

Q




micro j



micro j




References












946 R P Lipton












946 R P Lipton












Assessment of the local stress state through macroscopic variables

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