Popov, G.Osaka J. Math.24 (1987), 1-12

SOME ESTIMATES OF GREEN'S FUNCTIONSIN THE SHADOW

GEORGI POPOV

(Received July 30, 1985)

0. Introduction

The purpose of this work is to investigate the asymptotic behaviour ofGreen's functions in the so-called shadow for Laplace operator in an exteriordomain. As a consequence a field scattered by a non-trapping obstacle willbe examined at high frequencies.

These asymptotics have been studied by many authors since Keller'sarticle [6] appeared. It was shown that for some convex obstacles the scat-tered field in the shadow should be as small as the exponent exρ(—A\k\ 1 / 3),u4>0, is when the frequency k tends to infinity. Such an estimate is believedto take place for a large class of domains but it has not been proved yet even forstrictly convex obstacles except for some special cases. In [12], Ludwig con-structed an asymptotic solution UN for Helmholtz equation in the deep shadowwhich behaved like exp(—A\k\ 1 / 3 ), A>0, as &-»oo, but he did not show thatthe difference between UN and the exact solution could be estimated by thesame exponent.

The asymptotics of Green's functions in the shadow were investigatedin [1], [2], [3], [14]. Recently, an asymptotic solution of Green's functionsin the deep shadow was obtained by Zayaev and Philippov [4] for planar strict-ly convex obstacles. Probably, the technique developed in [8], [9], [11] maybe used to obtain the asymptotic expansions of Green's functions at high fre-quences for any strictly convex obstacle in Rn, n>2.

Let K be a compact in Rn, n>2, with a real analytic boundary Γ and let

Ω=Rn\K. The obstacle K is called non-trapping if for any R>0 with KdBR={x^R

n\ \x\Q such that there are no generalizedgeodesies, (for definition see [13]), with length TR within ΩΓ\BR. Denoteby Δ0, respectively by Δ^, the self-adjoint extension of the Laplace operatorin J?n, respectively in Ω with Dirichlet boundary conditions. Let

Rftk) = (-Δ,-*2)-1

be the resolvent of the operator — Δy, 7=0, D in ±Im&>0. Consider the

2 G. POPOV

cut-off resolvents

(0.1) {ΛeC; ±Im&>0} Ξϊk -> R^(k) = XRftk) %e_£(L2(Ω), L2(Ω))

where %eC(°5)(Π)— {9?eC°°(Π); supp^? is compact} and %(#)=! in a neigh-bourhood of Γ. Hereafter -C(Hly H2) stands for the Banach space of boundedlinear operators mapping from the Banach space /fj into the Banach space H2and equipped with the usual norm. Obviously the functions (0.1) are analyticwith respect to k in ±Im&>0.

Our first result is

Theorem 1. Suppose K non-trapping. Then the function (0.1) admitsan analytic continuation in the region

for some positive constants a and β.

This theorem was proved for strictly convex obstacles with C°° boundariesand for n=3 by Babich and Grigorieva [2]. Recently, in [8], [9], Bardos,Lebeau and Rauch showed that the region C/~β is free of poles of the scatteringmatrix for any non-trapping obstacle with an analytic boundary, providedn>3 odd. They investigated the generator B of the semi-group Z(t) intro-duced by Lax and Phillips in [7]. Using the propagation of the Gevrey sin-gularities of the solutions of the mixed problem for the wave equation theyproved the estimate \\BJ'Z(tQ)\\0 which is usually called outgoing (incoming)Green's function. For any k>0 the distribution G±(Λ, x,y) solves the problem

(0.2)

'(^+k2)G±(ky x, y) = -δ(x-y) , (x, y )€=ΩxΩ

^k, x, y) = Q(rW*) , =Fifc G± =dr

_as r = \x— j|->oo and k^R\ = (0,

SOME ESTIMATES OF GREEN'S FUNCTIONS 3

where B u=u/Γ.

The point #0eΩ belongs to the shadow Sh(y0) of K with respect to agiven point y0&Π if there are no generalized geodesies starting at j>0 and passing

through x0. Denote by d(x> y) the distance function in Ω, i.e.

d(x, y) = inf {length of 7; 7 is a path in Ω connecting x and y} .

Denote Z>J=D{ι— Z)J«, where Dy=i" 19/8Λ?yandί=(/)1, ,ίll)eZ;,Z+={0, !,-••}.

Theorem 2. Suppose K non-trapping and xQGSh(yQ). Then there existsa neighbourhood O of (xQ, yQ) inΩxΩ stick that

(0.3) \DfDpxD^yG±(ky x,y)\< C&φ(-A\k\^=fd(x9 y) Imk)

in U^βXO for any (m, p, q)^Z2++l and for some positive constants a, /3, -4, and

C=C(m,p,q).

Now consider the scattering of plane waves by the obstacle K. Lett'9 |0|=1} and denote Ls={χ(=R

n\ =*} where =

Consider the solution us(k, x) of the problem

(Δ+k2)us(k, Λ?) = 0

The point XQ belongs to the shadow Sh(K, ω) of K with respect to a given

direction ω if non of the generalized geodesies γ(ί), £>0, starting at Ls for some

ί

4 G. POPOV

estimated in [14] as follows

\G±(k,x,y)\0 in k>k0>0 and (#, y) in a neighbourhood of (#0, yQ).The estimate (0.4) was predicted by Keller's geometrical theory of dif-

fraction [5], [6], see also [12].The method we use is close to that developed by Vainberg [18] (see also

[16]) in order to prove uniform decay of the local energy for hyperbolic equa-tions. The propagation of Gevrey singularities for the mixed problem studiedin [10], [11] and the non-trapping condition allow us to compare the solutionsof the mixed problem with suitably chosen solutions of the Cauchy problemfor the wave equation. This is used in Proposition 1 to prove that the kernelsof the cut-oίϊ resolvents /Zjt3C(&) coincide with the Fourier transforms of somecompactly supported distributions modulo exponentially decreasing functions,holomorphic in C/Jβ. The theorems follow from Proposition 1 by using oncemore the results on the propagation of Gevrey singularities for the mixed prob-lem.

1. Estimates of Green's functions

In this section we prove theorems 1 and 2. Let us denote by U0(t) andU(t) the propagators of the Cauchy problem and the mixed problem respec-tively, i.e.

Γ(9?-Δ) [/„(*)/(*) = 0 in (ί, *)

I U0(Q)f(x) = 0 , dtU0(Q)f(x) =/(*) , /eCβ(Λ ) ,

( (9?-Δ) U(t)f(x) = 0 in (t,x)l> if forany compact M^M there exist some constants A=A(Mly /), B=B(Mly f)such that

for any α, \a\=al-\Let %eG3(Λn), X(x)=l in a neighbourhood of BR={x\ xR. In view of the non-trapping condition thereexists Γ>jR1 such that any generalized geodesic starting at BRι leaves it by

SOME ESTIMATES OF GREEN'S FUNCTIONS 5

the time T. Then from the theorem about the propagation of Gevrey Gz

singularities proved by G. Lebeau [10] follows that the distribution kernelU(ty xy y) of U(t) is a G

3 function in

Q0 = [R\(- T, T)] x (BKl Π Ω) x (BXί Π Ω) .

Therefore the estimate

(1.3) \

holds in (t, x, y)^Q for any compact QcQ0 and any y, α, β. Moreover theconstants AQ and CQ do not depend on (t, x9 y)^Q and onj, a, β.

Let ζt=G\Rn+l), ζ=l in a neighbourhood of the set {(ί, x)eR*+l; \\x\—t\ T+ 1 . Consider the operators

Next we write the modified resolvent ΛJ>X(&) in the form

(1.4) R+D^(k) = xέ(k)+Z^(k)

where

= Γ eikt XE(t)dt , Im k>0 ,o

denotes the Fourier-Laplace transform of %£'eL1(JR1, _£(L2(Ω), i2(Ω))). Note

that the operator-valued function %E(t) has a compact support with respectto t since X(x) ξ(t, x) has. Therefore XE(k) is an analytic function with valuesin the space jC(£2(Ω), £2(Ω)), while Z%(k) is analytic in {ΛeC; Im&>0}. Let/ίs(Ω), ί^O, ^e^, be the closure of C(°S)(Ω) with respect to the Sobolev norm\\u\\2s= Σ \\D«u\\

2

L2(Q) and let fl-f(Ω) be the dual space of /ίs(Ω). We shall use

\

G. POPOV

is the commutator of the operators F1 and F2 and ξ stands for the operator ofmultiplication by the function ζ(t, x). Then E(t) is the propagator of theproblem

(1.6)

The distribution kernel F(t, x, y) of the operator F(t) belongs to the Gevreyclass G3(R1xΏ,xΩ) in view of the propagation of Gevrey singularities ofU(t, x, y) and the definition of the functions f (ί, x) and %(#). Moreover

BE(t)f=Q

L £()/(*) = 0, 9,£(0)/(*) = %(*)/(*), /EΞL2(Ω).

(1.7)

\t\>T,T

SOME ESTIMATES OF GREEN'S FUNCTIONS

be continued analytically for Im ktϊ-*E(t}ϊΞj:(D-s, D~s+l) is bounded for any s0, s0y

, and

(1-10)

for anyIn view of (1.6), (1.7), (1.10) and Duhamel's formula we write

W(t)f(x) = M(x) E(t)f(x)+ Γ U0(t-s) G(s)f(x)ds,Jo

Note that the support of the distribution kernel of G(t) is contained in {(£, x, y)\t\

8 G. POPOV

According to (1.10), (1.11) and lemma 1 the function

[Γ2, oo)3ί -* %2W(t) eE_£(tf-*(Ω), fl*(Ω)) , T2 = 22\+2 ,

can be continued as an analytical one in {ίeC; \t\ >T2} for any t>0 and any%2e Co(BT\BR). Moreover the estimate

(1.13) l l

is valid in \t \ >Γ2 for any y>max(0, 3— n) and any s>0, $e2Γ where the con-stant A does not depend on j.

Now we can estimate the norm of the Fourier-Laplace transform ofin X(H"S, Hs). Let Re&>&0>0 for some A0>0. Since W(t)=U inwe can write

- &-1 Γ2 β'« DtX2W(t) dt+k-1 Γ eikt DtX2W(t) dt .

Jθ JΓ2

Using (1.13) we can change the contour of integration in the second integral toobtain

exp(C I k 1 1/3) Xjfr(k) = ΣC'\k\ y/3 -\j I)'1 [ 2 eikt DtX2W(t) dtj=o Jo

+e«r2 f" β-« χ^D )̂ (yι+ώ) Λ] .Jo

Integrating |j/3] times by parts in any member of the last sum we have

exp(C \k \ 1/3) %2T^(&) = f] Cy ̂ /3-[;73]-i( j)-ι

y=o

W(t) dt+eikT2 Γ e~kt %2(Z)[

ί

y/3Hi W) (T2+it) dt]JoJo Jo

where [m] denotes the integer part of m^R\. Since W&G3 and in view of(1.13) any member of the last sum can be estimated by

T whenlm&>0

T2 whenlm&£0>0}, where the constants A1 and Bl do not depend onProvided that C0, where C9=Al(l — CB\/3)~1. Proceeding in the same way

when ReA

SOME ESTIMATES OF GREEN'S FUNCTIONS

(1.15) l l l_/ : (H-mH*(Ω))^C. «p(-C|A| *-

is fulfilled in C\{k; lmk %.£(&) e_£(.ίΓ(Ω), ίΓ(Ω)) is analytic and

(1.16)

for any

Proof. The assertion is obvious for t=0 since U(t) is a bounded functionin Rl with values in -£(L2(Ω), L2(Ω)) and % f (ί, #)=0 for any t>2T+l. Sup-pose ί>l and consider

n i n Γ (Δ-(1>17) |_X£(

for/eίΓ(Ω). Here

and L(Λ) satisfies the estimate (1.16) for any s>0 since the distribution kernel

of the operator XF(t) is smooth and supp(XF) c { | x \

10 G. POPOV

IDfZHD ZΛ*, x,y)\ = \φpx δ,, D?ZX(Λ)Z>* ,

in U*tβXθ for some α>0 and A0>0. Here n+p-5rq. On the other hand ξ(t, x) U(t, x, y) is a G3 function in R1X Owith a compact support with respect to t. Moreover, U(t, x, y)=0 for \t\<

d(x, y) since the propagation speed for the solutions of the mixed problem for

the wave equation equals one (see [17]). Now the arguments used in the proof

of (1.14) yield (0.3).Denote by #(λ, x, y) the spectral function of the operator —Δp given as

the distribution kernel of the spectral projector Eλ of —Δp. Since £"λ—>/ inL2(Ω) asλ->oo and

de ^2 χ9 y) = (2n ί)-l{G+(\, x, y)-G-(\, x, y)} for x*y, λ>0 ,

d\ ^ ' *"

it is easy to obtain from theorem 2 the following

Corollary 1. Suppote K non-trapping and xQ^Sh(yQ). Then

IDjΓ DpxDq

y e(\9 x,y)\0 ,

in [X0, oo)χOfor (m, p, q)0.

2. Asymptotics of the scattered waves

In this section we prove theorem 3. Translating the origin to a givenpoint #0e/2

n the function us(ky x) is multiplied by exρ(/&

SOME ESTIMATES OF GREEN'S FUNCTIONS 11

, *) = %(*) Γ ξ(t-s, x) U(t-s) [Δ, φ] 8(s-0, y φfiΛ(Λ?0), rj=-τω, τφO} .

Moreover, theorem 1.4 in [10] yields

. {(φ'-^y, T, 7); f>y, T, ^); s 7ι(0 is tne generalized geodesic with initial data 7ι(0)=

7(0)—sω0 and 7 is a generalized

geodesic with γ(0)eL0, —^-(0) = ω} .dt

Since x^Sh(K, ω) we can choose a neighbourhood O of #0 such that (sing

*(vι)) Π 0—φ which proves theorem 3 since suρρ(^1) is compact.

12 G. POPOV

References

[1] V. Babich and V. Buldyrev: Asymptotic methods in short wave diffractionproblems, Nauka, Moscow, 1972 (in Russian).

[2] V. Babich and N. Grigorieva: Asymptotical properties of solutions of some threedimensional problems, Notes Sci. Sem. Steklov Math. Inst. (Leningrad Branch)127 (1975), 20-78 (in Russian).

[3] V. Philippov: On the exact justification of the short wave approximation in ashadow region, Notes Sci. Sem. Steklov Math. Inst. 34 (1973), 142-206 (in Russi-sian).

[4] A. Zayaev and V. Philippov: On exact justification of Friedlander-Keller formula,Notes Sci. Sem. Steklov Math. Inst. 140 (1984), 49-60 (in Russian).

[5] J. Keller: Geometrical theory of diffraction, J. Optical Soc. Amer. 58 (1962), 116-130.

[6] J. Keller: Diffraction by a convex cylinder, Trans. IRE, Ant. and Prop. 4, 3,

(1956).[7] P. Lax and R. Phillips: Scattering theory, Academic Press, 1967.[8] C. Bardos, G. Lebeau, J. Rauch: Estimation sur les poles de la diffusion, Meth-

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[9] C. Bardos, G. Lebeau, J. Rauch: Estimation sur les poles de la diffusion, pre-print.

[10] G. Lebeau: Propagation des singularίtes Gevrey pour le probleme de Dirichlet,Advances in microlocal analysis, NATO ASI Series, Ser. C, Math. Phys. Sci. 168(1985), 203-223.

[11] G. Lebeau: Regularite Gevrey 3 pour la diffraction, Comm. Partial DifferentialEquations 9 (1984), 1437-1497.

[12] D. Ludwig: Uniform asymptotic expansion of the field scattered by a convexobject at high frequencies, Comm. Pure Appl. Math. 20 (1967), 103-138.

[13] R. Melrose and J. Sjostrand: Singularities of boundary value problems, I, II,Comm. Pure Appl. Math. 31 (1978), 593-617 and 35 (1982), 129-158.

[14] G. Popov: Spectral asymptotics for elliptic second order differential operators,J. Math. Kyoto Univ. 25 (1985), 659-681.

[15] G. Popov and M. Shubin: Complete asymptotic expansion of the spectral functionfor elliptic second order differential operators in Rn, Functional Anal, i Pril. 17(1983), n°3, 37-45 (in Russian).

[16] J. Rauch: Asymptotic behaviour of solutions to hyperbolic partial differential equa-tions with zero speeds, Comm. Pure Appl. Math. 31 (1978), 431-480.

[17] J. Rauch: The leading wavefront for hyperbolic mixed problems, Bull. Soc. Roy.Sci. Liege 46 (1977), 156-161.

[18] B. Vainberg: On the short wave asymptotic behaviour of solutions of stationaryproblems and the asymptotic behaviour as £—>oo of solutions of nonstationary prob-lems, Russian Math. Surveys 30 (1975), 1-53.

Institute of MathematicsBulgarian Academy of Sciences1090, Sofia, Bulgaria