The Journal of Geometric Analysis Volume 4, Number 2, 1994

Weakly Monotone Functions

By Juan J. Manfredi

ABSTRACT. The definition of monotone function in the sense of Lebesgue is extended to the Sobolev spaces W j,p, p > n - 1. It is proven that such weakly monotone functions are continuous except in a singular set of p-capacity zero that is empty in the case p = n. Applications to the regularity of mappings with finite dilatation appearing in nonlinear elasticity theory are given.

1. Introduction.

Let [2 be a bounded domain in N n and f : ~ -+ ~ be a continuous function. We say that f is monotone if for every relatively compact subdomain ~2' of f2 we have that

sup f (x) = sup f (x) xEO~ ~ xE~2'

and

inf f(x) = inf f(x). xEO~' ~ xEf2'

Roughly speaking, mononote functions satisfy the maximum and minimum principles in s Exam- pies of monotone functions include solutions of homogeneous elliptic partial differential equations.

This definition of monotone function is due to Lebesgue (see [L] and [M]), who studied mono- tone functions in connection with the solution of the Dirichlet problem in the plane. The key feature used by Lebesgue is that a family of monotone functions with bounded Dirichlet integral is equicon- tinuous (see (2.5) below). We have recently extended a classical theorem of Lindeltif concerning asymptotic values and nontangential limits to a class of monotone functions [MV].

Math Subject Classification 35J70, 30C65. Key Words and Phrases Finite dilatation, monotone functions. Research partially supported by NSF grant DMS-9101864.

@ 1994 The Journal of Geometric Analysis ISSN 1050-6926

394 Juan J. Manfredi

For 1 < p < oo, let Wl~o~P(72) denote the local Sobolev space of functions in L~c(g2 ) whose distributional derivatives are in P Llo c (~) . Consider a mapping

F : 72---~ R ~

l,p in Wlo c (f2, ]~n). Denote by JF(X) the Jacobian determinant det(DF(x)) . We say that the mapping F hasfinite dilatation if

IDF(x ) I n K (x ) --

JF(X)

satisfies

1 < K(x ) 0 for a.e. x, then F has finite dilatation. In 1977 Vodopyanov and

I n Goldstein [VP] proved that a mapping F in Wlo ~ (f2, R n) with finite dilatation is indeed continuous and has monotone components. Their proof uses properties of the topological degree of a mapping in an essential way and shows that it can be defined in this case.

The adjugate matrix of DF(x) is denoted by adj(DF(x)) and it satisfies

DF(x) . adj (DF(x)) = JF(X). In (1.1)

If Jt~(x) ~s 0 we have adj( (DF(x)) ) = JF (X) (DF(x) ) -1 and, in general, the entries of adj(DF(x)) are homogeneous polynomials of degree n -- 1 with respect to the variables aFi/Oxj. In his fundamental work on nonlinear elasticity, Ball [B] introduced the classes

.A+,q(~) : {F 9 f2 --+ R", DF ~ LP(72), adj (DF(x)) ~ Lq(72) and JF(X) > 0 a.e.}

where p > 1 and q > p / (p -- 1). Note that by (1.1), these conditions guarantee that JF(X) E + L I (~) . In 1988, Sver~ik [S] proved that if p > n -- 1, mappings in the class .Ap,q ( ) are continuous

except in a set of p-capacity zero and this set is empty if p = n. Sverfik's proof again relies on the notion of topological degree and some results in geometric measure theory.

In this paper we propose an approach to these theorems that does not rely on the notion of topological degree. Roughly speaking, we extend the proof of continuity of quasiregular mappings

l,n to cover these situations. Quasiregular mappings are mappings in Wlo c (72, ~n) for which K (x) is bounded. It turns out that quasiregular mappings are continuous, open, and discrete [R].

The plan of the paper is as follows. In Section 2 we extend the notion of monotonicity to possibly noncontinuous functions in W:o~(f2) and prove that for p > n - 1 these generalized monotone functions are continuous except in a set of p-capacity zero. Moreover, this singular set is empty if p = n. We use the Sobolev Theorem on spheres and some tools from nonlinear potential theory. In Section 3 we consider a slight generalization of the classes .Ap+q (~'~) by dropping the requirement JF (x) > 0 a.e. and asking instead for finite dilatation. For p > n -- 1 and a large class of functions qJ,

Weakly Monotone Functions 395

including for example all harmonic functions, we prove that qJ ( F (x)) is monotone in the generalized sense and therefore continuous except in a set of p-capacity zero, which is empty if p = n. Our main tool here is the well-known fact that the columns of adj (DF(x) ) are divergence-free. In particular, it follows that log I F(x)l is monotone in the classical sense if p = n on f2 \F - l (0).

2. Weakly monotone functions

Definition. Let g2 be an open set in ~n and f ." f2 --+ R be a function in the space Wllo'cP (~). We say that f is weakly monotone if for every relatively compact subdomain ~' of g2 and for every pair of constants m < M such that

(m - f )+ E W~'P(f2 ')

and

( f - M) + E Wlo'P(f2'),

we have that

m < f (x ) < M fora.e, x E f2'. (2.1)

Since functions in Wl~c p (~) are p-quasi-continuous and two p-quasi-continuous functions that agree a.e., indeed agree except in a set of p-capacity zero, we see that (2.1) holds except in a set of p-capacity zero. For these results see for example [HKM].

If p > n then f is a continuous functions and if it is weakly monotone, then is monotone in the sense of Lebesgue. However if p < n is it not always true that weakly monotone functions are continuous.

Example 2.1. Write z = re i~ for z E ~x 2. Define a function f as follows:

f ( z ) =

0 for 0 < 0 < zr/2, n' /2 for 7r/2 < 0 < zr,

3zr/2 - 0 for zr < 0 < 37r/2, 0 for 3:rr/2 < 0 2re.

Then f E WILP(]~ 2) for any p < 2 is weakly monotone but not continuous at zero.

Theorem 1. Let f be weakly monotone in an open set f2 and p > n - 1. Then f is continuous in f2 \ S, where the singular set S has p-capacity zero. Moreover, if p = n, the singular set is empty, that is, f is continuous in ~.

396 Juan J. Manfredi

Proof. Fixx0 E f2andR > 0suchthatBR(x0) C f2. Aconsequenceofp-quasi-continuity for p > n -- 1 is that f is continuous on S(xo, r) (the sphere centered at x0 with radius r) for r E (0, R) \ E(xo, R), where the linear measure of E(xo, R) is zero. Pick an r ~ E(xo, R). We can then define

and

Mr(xo) = sup{f (x) : x e S(xo, r)}

mr(xo) = inf{f(x) : x ~ S(xo, r)}.

It is now easy to see (Chapter 1 in [HKM]) that (mr(Xo) - f)+ and (f -- Mr(xo)) + are both in WI'p(BR(Xo)). Thus, we conclude that indeed

mr(xo)

Weakly Monotone Functions 397

Note that S = 13 for p = n. We claim that Capp (S) -~- 0. It is enough to check that Capp (S fq f2') = 0 for a relatively compact subdomain g2' of Q. To check this define a function in 11~ n as follows:

w(x) = [Vf (x ) l e for x E f2' and 0 otherwise. Denote by l~(x) = Ixl ~-" the Riesz kernel of order or. We see that S 71 g2' is contained in

H = {x : (I e* w)(x) = +oo}.

It then follows that any compact subset K of H must have Riesz capacity of order p equal to zero (see Theorem 5.9 in [HK]). Thus we must also have caPe(K) = 0. To conclude that cape(H) = 0, all that is needed is the capacitability of H. But since w is nonnegative, one can easily check that Ip * w is lower semicontinuous and therefore H is a Borel set.

Note that for r ~ (0, R) \ E(x0, R) we have

ess-oscB9 = OSCS(xo,r) f .

Therefore, we have established that if, in addition, Xo ~ S, then

eSS-OSCBr(xo) f < C(n, p) log aS(~o,t) " (2.5)

Now define

g(xo) = lim f f (y ) dy. r~O JBr (xo)

Note that for r ~ (0, R) \ E(xo, R) we have

< f f (y ) dy < mr(Xo) mr(xo), dB ,(xo)

and by (2.5)

Mr(xo) - mr(XO) --+ 0

as r --+ 0 in (0, R) \ E(xo, R) whenever x0 S. We conclude that the limit defining g(xo) always exists for x0 ~ S if r --+ 0 restricted to (0, R) \ E(xo, R). Since for fixed x0, the function

f f (y ) dy r JB 9 (xo)

is a continuous function of positive r, the limit defining g(xo) exists also as r --+ 0. Moreover, we obtain

lim mr(xo) = lim Mr(xo) = g(xo) for x0 ~ S. (2.6) r---~O r---~O

398 Juan J. Manfredi

Clearly f (x ) = g(x) for a.e.x. We wish to prove that g is continuous in f2 \ S. Using (2.6) we get

ess- lim inf g(y) = ess- lim inf f (y ) y--~ xo y--'~ xo

= limmr(Xo) r---~O

= g(Xo) .

Therefore g satisfies for x0 r S the condition

g(xo) = ess- lim infg(y). (2.7) y ~ xo

It is now an elementary exercise to show that (2.7) implies that g is lower semicontinuous in f2 \ S. A symmetric argument gives that g is upper semicontinuous in f2 \ S since we also have

g(xo) = ess- lim sup g(y). [] y---~ Xo

Coro l lary 1. Weakly monotone functions are always locally bounded for p > n - 1.

Proof. It is enough to consider the case where n - 1 < p, p ~ n. Direct integration of (2.3) gives the following oscillation estimate for weakly monotone functions in W l'p (~):

iV f (y)lP dy (ess-osc(f, B(xo, r))) p < c(n, p)r p "(x0,2r) (2.8)

whenever B"(xo, 2r) C f2. [ ]

Remark 2.1. Note that (2.8) also holds for p = n. Indeed (2.5) is a better inequality in this case, since the continuity of f implies that the S = 0 and E(xo, R) = ~. []

Coro l lary 2. Suppose that u E WI 'P (R n) is weakly monotone and p > n - 1. Then u is identically zero.

Proof. It follows by letting r ~ oo in (2.8). [ ]

Weakly Monotone Functions 399

3. Mappings of finite dilatation

wl ,n - I t 'O Let ~ and ~2' be domains in R" and F: g2 ~ ~' amappingin ,,lo~ ~,,, R"). It is a well-known fact that adj (D F) satisfies

div(adj(DF)) = 0

in the sense of distributions. We shall need a generalization of this fact that can be found in [BI, Corollary 1.1]. Let V: f2' ~/1~" be a C1(f2 ') vector field such that div V = 0. Then we have

d iv (ad j (DF) (V o F)) = 0 (3.1)

in the sense of distributions.

Note that the hypothesis on F guarantees that adj (DF) E L~o~(~2). A simple approximation argument gives that if ad j (DF) E Lqoc(f2), where q > 1, then

a Iad j (Df (x ) )V(F (x ) ) , V~b(x))dx = 0 (3.2)

for all q~ E w~'q'(fl). Here q' = q/ (q -- 1) is the conjugate exponent ofq.

We now present a generalization of the classes .A+,q(f2). For p > 1 and q > p / (p -- 1) we consider

B+q(~) = {F : ~2 -+ 1R ~, DE ~ LP(~) , ad j (DF(x ) ) E Lq(f2)

and f has finite dilatation a.e. in f2}.

Theorem 2. Let F: f2 --~ f2' be in the class ]3~q ( ~'2 ), where p > n -- 1 and q > p / (p - 1). Suppose that V E C1(~ ', ~n) satisfies div(V) = 0 and cb: ~ ' --~ R is a Cl function such that

V~(y) = )~(y)V(y) fora.e, y E ~ ' , (3.3)

where 0 < )~(y ) < cx~ a.e. in ~2' and )~ is bounded on compact subsets. The function ~ ( F (x ) ) is then weakly monotone.

Before proceeding with the proof we will give some examples. If the function ~. and the range f2' are not explicitly given below, ~(y) =--- 1 and f2' = Nn.

....+ Example 3.1. Let V(y) = e i and qb(y) = Yi. We conclude that the components of

F, F l (x ) , F2(x) . . . . . Fn (x ) are weakly monotone. By Theorem 1, they are continuous except in a set of p-capacity zero, which is empty if p = n. In particular, we recover the theorems of Vodopyanov-Goldstein and ~ver~ik mentioned in the introduction.

400 Juan J. Manfredi

Example3 .2 . Let V(y) = y/ ly l n, ~(y) = log [YI,~-(Y) : lyln-2, ands ' ~- ]~'\Bs(O). We conclude that log [F(x) l is weakly monotone if F does not vanish in the sense that [F(x)[ > for x E ~. In these circumstances, an oscillation estimate for log [F(x)l is readily translated into a Harnack inequality for [F(x)l. If follows from (2.8) that if BR(xo) C ~ and we set

Mr = max{ IF (x ) l : x E Br(xo) },

mr = min{ lF (x ) l " x c Br(xo) }

we must have for 0 < r < R/2

M, < ci exp {c~ (fBR(xo) ,Vlog,F, lP dx) }

where cl and c2 are constants depending only on n and p.

mr,

""+ ---+ y2 2, Example 3.3. Let V(y) = Yi e i - Yj e j, O(y) = i - Y'j and ~,(y) ---- 1/2. We conclude that (F i (x)) 2 - (F j (x)) 2 is weakly monotone.

Example 3.4. Let V(y) = Yi e j + yj e i, i 5/= j , and q~(y) = YiYj. We conclude that F i (x) F j (x) is weakly monotone if i 5~ j .

Example 3.5. This example generalizes the previous ones. Let qb be any harmonic function in f2'. Take V(y) = V~(y) to conclude that

9 (F l (x) . F2(x) . . . . . F"(x))

is weakly monotone.

Example 3.6. This example is even more general. Let ~ be any r-harmonic function for some 1 < r < oo, that is, qb satisfies the r-Laplace equation

d iv ( lV~l r -2v~) = O.

Set V (y) = IV 9 (y) it-2 7 q~ (y), which is smooth in the open set

~' = {y :o < IV~(y) l < oo}.

Weakly Monotone Functions

Taking X(y) = I V~ (y)I r-2 we conclude that

qb(F1 (X), F2(x ) . . . . . Fn(x ) )

is weakly monotone.

401

Proof of Theorem 2. Set r = dp(F(x)). Then, by the chain rule we have

Vr : (DF(x))tVdP(F(x)) = )~(F(x))(OF(x))tV(F(x)).

Suppose that ( r - M) + E Wd'P(f2 ") for some constant M, where f2" is compactly contained in

f2. It is enough to prove that

r < M for a.e. x E f2" (3.4)

Since p > q' we can use (r - M) + as a test function in (3.2) to obtain

f~ (adj(OF(x))V(F(x)), ;~(F(X))(DF(x))tV(F(x))) dx 0 ----- "nlr

= f~ )~(F(X))lV(F(x))12JF(x)dx" "n{r

Since the integrand is nonnegative, we conclude that IV(F(x))12 = 0 for a.e. x in

n = n {r > M} n {JF > 0}.

By virtue of (3.3) we conclude that Vr = 0 in H. Indeed, Vr = 0 in

gz"n {r > M}

since F has finite dilatation. We deduce that V ( r - M) + = 0 in ~", thereby proving (3.4). []

[B]

[BI]

[G] [HK] [HKM]

[L]

References

Ball, J. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63, 337-403 (1978). Bojarski, B., and Iwaniec, T. Analytical foundations of the theory of quasiconformal mappings in I~ n . Ann. Acad. Sci. Fenn. Sen A I Math. 8, 257-324 (1983). Gehring, E Rings and quasiconformal mappings in space. Trans. Amer. Math. Soc. 101,499-519 (1961). Hayman, W., and Kennedy, P. Subharmonic Functions, Academic Press, 1976. Heinonen, J., Kilpeliiinen, T., and Martio, O. Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, 1993. Lebesgue, H. Sur le probl~me de Dirichlet. Rend. Circ. Palermo 27, 371--402 (1907).

402

[MI

[MV] [R] Is]

[VP]

Juan J. Manfredi

Mostow, G. Quasiconformal mappings in n-space and the rigidity of hyperbolic space forms. Publ. Math. Inst. Hautes E, tudes Sci. 34, 53-104 ( 1968 ). Manfredi, J., and Villamor, E. Traces of monotone Sobolev functions. Journal of Geometric Analysis, to appear. Re~etnyak, J. Space mappings with bounded distortion. Sibirisk. Mat. Z. 8, 629-658 (1967). Sver~ik, V. Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal. 100, 105-127 (1988). Vodopyanov, S., and Goldstein, V. Quasiconformal mappings and spaces of functions with generalized first derivatives. Siberian Math. J. 17(3), 515-531 (1977).

Received February 18, 1992

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260; manfredi+@pitt.edu

Communicated by David Kinderlehrer