low frequency asymptotics for the reduced wave equation in two-dimensional exterior spaces

Math. Meth. in the Appl. Sci. 8 (1986) 134-156 AMS subject classification: 31 A 25, 35 B 40, 35 J 05, 45 B 05

0 1986 B. 0. Teubner Stuttgart

Low Frequency Asymptotics for the Reduced Wave Equation in Two-dimensional Exterior Spaces

P. Werner, Stuttgart

We consider the Dirichlet problem for the reduced wave equation A Ux + x2 Ux = 0 in a two-dimensional exterior domain with boundary C, where C consists of a finite number of smooth closed curves C, , . . . , C,,, . The question of interest is the behavior of U, as x + 0. We show that U converges to the solution of the corresponding exterior Dirichlet problem of potential theory if the boundary data converge to a limit uniformly on C. This generalizes a well-known result of R. C. MacCamy for the case m = 1.

1 Introduction

In a recent paper [8] the asymptotics of the solutions of boundary and initial value problems for the wave equation and the heat equation with time- independent right-hand sides in two-dimensional exterior domains as t + m have been studied. The methods in 181, as far as the Dirichlet boundary condition is concerned, are restricted to the case that the boundary consists of a single closed curve. This restriction is not natural, so that it seems to be desirable to extend the analysis in [8] to the case of several boundary components C, , . . . , C,,, . In the following we shall describe a method which provides the desired extension. As the spectral-theoretical discussion in 181 shows, the asymptotics of the time- dependent problems as t + m are closely connected with the low frequency asymptotics for the reduced wave equation, which will be the central topic of this paper.

The investigation of the asymptotic behavior of solutions of the reduced wave equation A U + x2U = 0 as x -, 0 has a long history. The first general existence proofs for the exterior Dirichlet problem, due to W. D. Kupradse, H. Weyl and C. Milller, were based on integral equations which failed to be uniquely solvable for a countable set of real wave numbers x , including x = 0. The main step of the classical argument requires to modify the integral equations at the exceptional values of x in such a way that the orthogonality conditions in the second part of Fredholm’s alternative theorem are satisfied. Later efforts were directed towards the reduction of exterior boundary value problems to integral equations which are uniquely solvable for all wave numbers x with Imx 2 0. We mention the combined use of a double layer and a volume poten-

Low Frequency Asymptotics for the Reduced Wave Equation 135

tial, proposed in [7], and the combination of a double and a single layer potential, coupled by a constant imaginary factor (see, for example, [l]). A systematic account of the last method and later variations, due to several authors, is con- tained in (21.

The reduction of exterior boundary value problems to uniquely solvable integral equations provided a relatively easy way for the discussion of dependence properties of the solutions. For example, it has been shown in [7] for the exterior boundary value problems in three dimensions that continuous dependence of the data on x implies continuous dependence of the solutions on x if Imx 2 0. The methods in [7] can be easily extended to space dimensions n > 3. In particular, the analysis in 171 implies that the solution of the exterior Dirichlet problem for A u + x2u = 0, subject to Sommerfeld’s radiation condition, converges for every space dimension n > 3 to the solution of the corresponding exterior Dirichlet problem of potential theory as x + 0 if the boundary data yx converge uniformly to a limit yo.

An extension of the methods in [7] and [l] to the smallest dimension n = 2 leads to difficulties, since the fundamental solution H&’)(x I x - y I ) of the reduced wave equation in two dimensions, and hence the corresponding volume and single layer potentials, have no limit as x + 0. The earliest results in two dimensions are due to R. C. MacCamy, who studied in [5 ] the low frequency behavior of the solution of the exterior Dirichlet problem in the case n = 2, by using only a double layer potential. This approach leads to an integral equation which is not uniquely solvable in the limit case x = 0. MacCamy proved by a very interesting singular perturbation argument that the solution converges to the corresponding potential-theoretical solution also for n = 2 under the additional hypothesis that the complement of the exterior domain is connected. This excludes the case that the boundary C consists of several connected components Cl , . . . , C,. The case rn = 1, considered by MacCamy, is distinguished by the fact that the homogeneous integral equation, resulting for x = 0, admits only one linearly independent solution.

Motivated by MacCamy’s singular perturbation argument, R. Kress has systematically studied families of compact operator equations (I - A,)x, = f, with the property that (I - Ax)-’ exists for x * 0 and the null space of I - A. has arbitrary (finite) dimension. In particular, he described in [3] the asymptotic behavior of x, as x -+ 0 under suitable assumptions on A, and f, . Furthermore, he applied the abstract theory, developed by him, to the integral equation, resulting from the classical double potential method in the case of the exterior Dirichlet problem in three dimensions, and thus gave a new proof of some of the dependence properties, derived in [7] by the combined double layer and volume potential method. However, his remark that this extends MacCamy’s results to the case of several boundary components does not seem to be fully justified, since the two-dimensional situation, studied by MacCamy, differs essentially from the three-dimensional case, according to the singularity of the fundamental solution at x = 0. Thus the discussion of the asymptotic behavior of the solution of the exterior Dirichlet problem as x -, 0 seems to be still open in the case n = 2, m > 1. It is one of the aims of this paper to close this gap.

136 P. Wnner

Another method for the investigation of the limit x -+ 0 in the case n = 2, m = 1 has been recently proposed in [8]. The logarithmic singularity of the Hankel function HI) suggests to use a coupling factor, which varies with x, and to base the discussion of the low frequency behavior on the representation

The boundary condition U, = yx on C leads to the integral equation p, + Lxpx = yx with

Since

where y denotes the Euler-Mascheroni number, the operator family L, can be continuously extended onto x = 0, by setting

It has been shown in [8] under the assumption m = 1 that the only continuous solution of the homogeneous equation p + Lop = 0 is p = 0 (compare [8], Lemma 4.1). This remark yields a simple proof of MacCamy's convergence result for the exterior Dirichlet problem in the case n = 2, m = 1 (compare 181, Lemma 5.1).

An immediate extension of this method to the case m > 1 is not possible, since the integral equation p + Lop = 0 has non-trivial solutions for m > 1, as we shall show in section 2. Nevertheless, the existence of (1 + Lo)-' for m = 1 can be applied also to the discussion of the limit x -* 0 in the case m > 1, by pro- ceeding in two steps as follows: Denote the exterior of C = C, + . . . + C, by D and the exterior of the first boundary component Cl by D1. At first we consider Green's function Gx(x, y ) of the exterior Dirichlet problem for A U + x2 U = 0 in D1 and investigate the asymptotic behavior of G,(x, y ) as x -* 0, by using the already established dependence theory for m = 1. In particular, section 3 con- tains a new proof of the fact that G,(x ,y ) converges to the corresponding Green's function Go& y ) for the potential equation A U = 0 in D1 as x -* 0. This convergence property, which has been already obtained by L. A. Muravei [6] by different methods (see below), is quite remarkable, since it is not shared by the fundamental solution Hh')(x I x - y I ) of A U + x2 U = 0 in the whole two- dimensional space.

(1.5) G,(x, y ) -* Go(x, y ) as x -* 0 I

In the second ,step we apply the limit relation


to the Dirichlet problem

A U, + x 2 U , = 0 in D , Ux = yx o n C ,

u, = o(r-1/2),

for the exterior D of C = Cl + . . U i be the solution of the related problem

(1.6) (+ - ix) u = o(r-1’2) as r = 1x1- m

AUA + x 2 U i = 0 i n D , , Ui = yx o n C , ,

1 + C,,, with m > 1 and Imx 2 0, x =t= 0. Let

(+ - ix) ui = o(r-1’2) a s r = / X I + m .

Assume that yx -+ yo uniformly on C. By the dependence theory for m = 1, we have Ui --.) UA as x + 0, where Uh is the solution of the exterior Dirichlet problem of potential theory in D, with boundary data yo on C1 . Set C’ : = C - C1 = C2 + . . . C,. The investigation of the behavior of U, as x -+ 0 can be based on the representation

Note that U, = Ui = yx on CI . In order to satisfy the required boundary condition also on the remaining part C’ of C, we have to choose v, such that

(1.9) v,(x) + 2 5 V,(Y) ( - + 1) G,(x, y > b Y = y,(x) - ui(x) forxE c’.

We shall show in section 4 that the corresponding homogeneous equation has only the solution v = 0 in the limit case x = 0. It is easy to deduce from this fact and (1.5) that U, converges to the uniquely determined solution Uo of the Dirichlet problem of potential theory as x 4 0:

C‘

A U o = O i n D , uo = Yo o n C = C1 + . . . + C,, i U, = o(I) , OW, = o(re2) as r = 1 x 1 4 03 ’ (1.10)

Thus the results of MacCamy can be extended to the case m > 1. Furthermore, we shall prove that the limit relation (1.5) for Green’s function and the results in [8], section 4 on the asymptotic behavior of the solutions of the exterior Dirichlet problems for the wave equation and the heat equation as t + m remain valid for m > 1. The discussion of the time-dependent problems requires a precise estimate for the remainder U,’ - Uo.

Note that similar convergence properties do not hold for the exterior Neumann problem in two dimensions, as the argument in [8], section 3 shows (compare also 151). In contrast to the Dirichlet case, the discussion of the Neu- mann problem in 151 and 18) does not require any modifications for rn > 1. We

138 P.Werner

shall supplement the theory of the Neumann problem with a discussion of the low frequency behavior of areen’s function in the Neumann case at the end of section 4.

The low frequency behavior of Green’s functions of various exterior boundary value problems for the two-dimensional reduced wave equation has been studied in a series of papers by L. A. Muravei (compare [6] and the references given there). As mentioned above, he has shown by a different method that Green’s function of the Dirichlet problem converges as x + 0 if m = 1. The argument in [6] is restricted to the case m = 1 and uses the double potential method, as in MacCamy’s approach. In particular, the resulting integral equation is not uniquely solvable in the limit case x = 0, in contrast to the integral equations considered in this paper. Finally, we remark that A. G. Ramm has recently proposed different methods for the investigation of the behaviour of the solutions to exterior boundary value problems at low frequencies 191. His work includes the Robin problem and the Schrddinger equation with a potential of compact support.

2 Remarks on the exterior Dirichlet problem of potential theory

Consider m closed curves C, , , . . , C,,, in the plane and denote the interior of Cj by Dij . We assume throughout this paper that C, , . . . , C,,, are twice continuously differentiable and that n % = 0 f o r j * k. We set C := C, + - . - + C,,,, Di := Di, u . . . u Dim and denote the exterior of C by D. We choose the orientation of the unit normal vector n on ’C such that n points into the exterior D.

We consider in this section the family of integral equations

depending on a complex parameter a, and show: Theorem 2.1 constant functions (2.2) where yl , . . . , ym are complex numbers with the property

a =k 0, then the continuous solutions of (2.1) are the piecewise

p(x) = y,forxEC,(j = 1,. . .,m),

j - 1

(11 Gill := length of Ci). I f a = 0, then the continuous solutions of (2.1) are given by (2.2) with arbitrary complex numbers yl,. . . , y,,, (without the restriction (2.3)). In particular, the linear space, consisting of all continuous solutions of (2.1), has dimension m - 1 for a * 0 and dimension m for a = 0 . P r o o f . Let p be a continuous solution of (2 .1) and set

(2.4) V(x) := R C

Low Frequency Asymptotics for the Reduccd Wave Equation 139

for x E D u D,. It follows from standard theorems on double layer potentials that U can be extended to functions U, and U,, which are continuously differentiable in D and in c, respectively. Furthermore, we have A U = 0 in D u D,, U, = 0 on C, and U = O(l) , V I/ = O ( r - 2 ) as r = ( X I .+ UJ. Hence Green’s formula implies that

- au j u-&= j / v u ( ~ ~ x = o ( ~ - ’ ) a s r - + a o , i x ! - r a r X E D

1x1 < r

so that V U vanishes in D. Since normal derivatives of double layer potentials are continuous across C , i t follows that ( a / a n ) U, = 0 on C. By applying Green’s formula to the interior D,, of C, for j = 1 , . . . , m , we conclude that V U vanishes also in the interior D, of C. Hence U is constant in the intzrior D,, of each curve C,. This implies that p = (U, - U,)/2 = - U,/2 is constant on each C,. Thus p has the form (2.2).

In order to complete the proof of Theorem 2.1, we have to determine all functions of the form (2.2) that are solutions of (2.1). Let p be given by (2.2). Then we have for x E C,

. .

Denote the exterior of Cj by 0,. It follows from Green’s formula, applied to the interior D,, of C,, and the jump relation for double layer potentials that

, 0 fo rxEDj ,

-1 f o r x E C , . i a 1 (2 .6 ) - j -In- n C, any I X -YI

Hence (2.5) reduces to

By (2.7), p is a solution of (2.1) if and only if either a = 0 or (2.3) holds. This completes the proof of Theorem 2.1.

Now assume that m = 1 and let yo be a given continuous function on C = C, . It follows from Theorem 2.1 by the first part of Fredholm’s alternative theorem that the integral equation

1 (2.8) P ( X ) + - SPlCY)

n c

140 P.Werner

has a uniquely determined solution p for every a t 0. By the jump relation for double layer potentials, the function U, defined by (2.4)’ is the solution of problem (1.10). Thus Theorem 2.1 provides a very simple approach to the exterior Dirichlet problem of potential theory in the case m = 1. The traditional approach corresponds to the choice a = 0 and employs the second part of Fredholm’s alternative also for m = 1.

Furthermore, it follows from Theorem 2.1 that the integral equation (2.8), which results from the representation (2.4)’ is not uniquely solvable for m > 1. It is the main purpose of the following considerations to reduce the exterior Dirichlet problem (1.6) also for rn > 1 to an integral equation, which is uniquely solvable in the limit case x = 0 and thus can be applied to the investigation of the asymptotic behavior of the solution (I, of (1.6) as x + 0. As pointed out in the introduction, our strategy will be to use the results in the special case m = 1, in particular the existence of the inverse operator ( I + Lo)-’, as a bridgehead for the attack of the general case m > 1 by means of a Green’s function argument.

3 The low frequency behavior of Green’s function for m = 1

We assume in this section that m = 1. Let Lo be the integral operator introduced in (1.4). The inverse operator ( I + Lo)-’ exists by Theorem 2.1, since m = 1. By (1.2) we have

with respect to the operator norm (3.1) llL, - Loll- 0 ~ S X -+ 0

(3.2) IILII:= supr{II~~II:IbII = Ij . , lbIl:= maxlc((x>l. XE c

Hence the existence of ( I + Lo)-’ implies that also (Z + Lx)-’ exists for sufficiently small I x I, say for Ix I < @ with 0 < @ < 1, as a local Neumann expansion shows.

Green’s function G,(x, y ) of the exterior Dirichlet problem for the reduced wave equation A U + xz CJ = 0 describes the time-harmonic wave field, which results from the reflection of the two-dimensional point source (i/4)Hb’)(x Ix - y I), located at the pointy E D, with regard to the reflection condition U = 0 on C. The discussion of the behavior of Gx(xJ y) as x + 0 will be based on the representation

with (3.3) Gx (x, y) = Qx (XJ y) Rx (XJ u)

(11 C 11 : = length of C) and


for x, y E D, x * y, 0 < I x 1 c Q and Imx 2 0, where P,( . , y ) is the solution of the integral equation

Note that Q,(. , y ) is continuous on C for every y E D. By (1.3), Q,(x, y ) converges to

as x --* 0. More precisely, the estimate

(3.8) Q,(x,Y) = Q o ( x , ~ ) + 0 ( 1x1 2 In- a s % + 0

holds uniformly in Ml x M2 for every pair of disjoint compact subsets Ml , M2 of D.

G,(x, y ) has the following properties: 1 a) M,(x, y ) := G,(x, y ) - - H t ) ( x I x - y 1) has continuous derivatives of 4

arbitrary order in D x D; b)(A, + x2)G,(x, y ) = Oifx, y E D a n d x =k y ;

c) G,(x, y ) = o(r-1/2), ( - :x - ix) G,(x, y ) = o(r-1/2)as r = I x I + oo uni-

formly with respect t o y in every compact subset of D; d) G,(x, y ) -+ 0 as x + xo for xo E C uniformly with respect t oy in every compact subset of D; e) properties b), c), d) hold also for the derivatives (a/i3yj)G,(x, y).

In order to verify a), note that formulas (3.3) - (3.6) imply 1

(3.9) M A X , u) = R,(x, v) - - I H P ( x l x - Z l W , 411CIl c

with

(3.10) P,(. , Y ) = - (1 + U-' Q,(. , U) , Since ( I + L,)-' is bounded with regard to the operator norm (3 .2) , we obtain

(3.11) D,P,(.,Y) = - ( I + U - l D y Q x ( . , ~ ) for every differential operator D,, = (a/ay,)"I (a/ay2)u2, and D,p,(z, y ) depends continuously on z and y in C x D. This, together with (3.5) and (3.9), implies a). In particular, we have

142 ~ . w e r n e r

for x, y E D. By (3.3)-(3.5) and (3.12), property b) holds for G,(x, y ) and all derivatives D,G,(x, y) . Since p,(z, y ) and D,,p,(z, y ) are bounded in C x K for every compact subset K of D, it follows from (3.3)- ( 3 . 3 , (3.12) and the well- known classical formulas 2(d/dz)Hf) = HvLl - HL1il and

that c) is satisfied for G,(x, y ) and (8 /8y j )G , (x , y) . Furthermore, formulas (3.9), (3.5) and (3.12) imply, by the jump relation

= - - H p ( x ] x o 1 - Y l ) 4

as x -+ xo E C . It follows from (3.14), (3.11) and (3.4) by the same argument that

as x -+ xo E C. The estimates, used in the classical proof of the jump relation (see, for example, [2], p. 48-50), show that (3.13) and (3.14), and hence (3.15) and (3.16), hold uniformly with respect to y in every compact subset K of D, since D,,&(z, y ) is continuous in C x K. Thus d) holds for G,(x, y ) := (i/4)Hb1)(x I x - y I) + M , ( x , y ) and for every derivative D,G,(x, y) . This concludes the verification of the properties a) - e).

It follows from (3.1), by using the local Neumann expansion 0

(3.17) (I + L,)-’ = C (- l)’[(I + Lo)-’(L, - LO)]’(I + Lo)-’ , j - 0

that ( I + LJ-’ converges to ( I + LO)-’ as x -+ 0 with respect to the operator norm (3.2). Hence (3.8) and (3.10) imply that p,(. , y ) converges to

(3.18) P O ( . , Y ) = - ( I + Lo)- ’Qo( . ,y> as x -+ 0 uniformly in C x K for every compact subset K of D. This, in connection with (3 .3 ) - (3 .3 , shows that G,(x, y ) converges to

(3.19) GO(X U ) = Q o k Y ) + Y )

Low Frequency Asyrnptotics for the Reduced Wave Equation 143

as x -, 0 for all x, y E D with x #= y , where Qo and Ro are given by (3.7) and

an,. (X - ZI 1

(3.20) Ro(x,y) :=- Ipo(z,y) n c

Furthermore, the convergence is uniform in Kl x K2 for every pair of disjoint compact subsets Kl , K2 of D. The limit Go(x, y ) has the following properties, analogous to a) - e):

has continuous derivatives of arbitrary order in D x D; b,) A,Go(x, y ) = 0 if x, y E D and x + y ; c,) Go@, y ) = 0(1), V''G0(x, y ) = O(r-2) as r = 1x1 -, 00 uniformly with respect to y in every compact subset of D ; do) G,(x, y ) -+ 0 as x -+ xo for x, E C uniformly with respect t oy in every compact subset of D; e,) properties bo), c,), 4) hold also for all derivatives DyGo(x, y ) .

The verification of these properties is based on formulas (3.18) - (3.20) and (3.7) and proceeds in the same way as the proof of properties a) - e) above. In particular, c,) follows from (3.19). (3.20) and (3.7), by using the estimates

1

1 X - 7 I Mo(x,Y) := Go(x,Y) - - In

2n

and

as r = 1x1 -+ 00 and by observing that po(z, y ) is continuous, and hence bounded, in C x K for every compact subset K of D by (3.18). Note that c,) holds also for all derivatives DyCo(x, y ) , since also Dypo(. , y ) = - ( I + Lo)-'DYQo(. , y ) is continuous in C x K. By %) - 4), G,(x, y ) can be considered as Green's function of the two-dimensional exterior Dirichlet problem for the potential equation A U = 0.

In view of the applications to the time-dependent theory (compare [8]), a precise estimate of the remainder GJx, y ) - G,(x, y ) as x -+ 0 is required. By (1.2) - (1.4), we have

1 lnx

(3.21) L, - Lo = - - S + T, with

Ix - 4 1 (3.22) (Sp)(x) := - j p ( z ) n c

and

144 ~ . w c r n c r

By inserting (3.21) - (3.23) into (3.17), we obtain the asymptotic expansion

with

as x + 0 uniformly in C x K for every compact subset K of D. Thus we have shown that

(3.26) px(z, Y ) = po(z, y ) + - e ( z Y ) + 0 lnx

uniformly in C x K for every compact subset K of D, where e is given by (3.27) Q ( . , Y ) : = - ( I + LO)-'S(I+ Lo)-'Q0(.,y). Note that @ (2, y ) and all derivatives D,,e(z, y ) are continuous in C x D, since ( I + Lo)-' and S are bounded with respect to the norm (3.2). By using (3.11) and replacing Qo(. , y) by DyQo(. , y) in the above calculation, we obtain

uniformly in C x K for every compact subset Kof D, since Dy commutes with the bounded operators ( I + Lo)-' and S.

Now we insert the estimates (3.26) and (3.28) into (3.5). By observing (1.3) and (3.20). we obtain

- . r

1 (3.29)


with

and a similar estimate for the derivatives of RJx, y ) with respect to y :

Both estimates hold uniformly in K, x K2 for every pair of compact subsets K1, K2 of D. Note that

by (3.4) and (3.7). It follows from (3.3), (3.19) and the estimates (3 .8) , (3.29), (3.31) and (3.32) that

1 In x

(3.33) Gx(x, y ) = Go(x,y) + - A ( x , y ) + 0

and

uniformly in K , x K2 for every pair of disjoint compact subsets Kl , K2 of D.

case m > 1 in the next section. The estimates (3.33) and (3.34) are the main tools for the discussion of the

4 The low frequency behavior for m > 1

In the following we assume that rn > 1. We want to study the behavior of the solution Ux of the Dirichlet problem (1.6) as x + 0, where yx is a family of continuous functions, defined on C, with y x -, yo as x -+ 0 uniformly on C. Let Ui be the solution of the Dirichlet problem (1.7) for the exterior D, of the first boundary component C, and consider Green’s function Gx(x, y ) of the exterior Dirichlet problem for the equation A U + x2 U = 0 in D1 . We try to represent Ux in the form (1.8) with a suitable continuous functionpx, defined on C‘ := C - C, = C2 + . . . + C,,, . By (1 -8) and the properties (a) - (e) of GJx, y), collected in section 3 , Ux is a solution of A U, + x2 Ux = 0 in D C D1 and satisfies Sommer- feld’s radiation condition

146 ~.werner

if Imx 2 0 and x =k 0. Furthermore, Ux can be extended to a continuous function in fi and satisfies the required boundary condition Ux = Ui = yx on Cl . By the jump relation for double layer potentials and property (a), the boundary condition Ux = y, on the remaining part C’ of C is equivalent to the integral equation (1.9). By introducing the integral operator

(4.2) (PXv)(x):= 2 j v(y) ( - + 1) G&, y)dsY f0rx.s C‘ , C’

equation (1.9) can be written as

(4.3) Note that definition (4.2) can be extended to x = 0, by replacing G,(x, y ) by Green’s function Go(x, y ) of the Dirichlet problem for A U = 0 in the exterior D, of Cl . The estimates (3.33) and (3.34) imply that

v, + Pxv, = yx - u:.

where the norm is defined by

(4.5) IIPII:= supr{(lPvll’:Ilvll’ = I}, ~ ~ v ~ ~ ‘ : = maxlv(x)l

in analogy to (3.2). We show:

Lemma 4.1 The only continuous function on C’, satisfying v + Pov = 0, is v = 0.

P r o o f . Let v be a continuous solution of v + P o v = 0 and set

n C ’

(4.6) V(X):= 2 j ~ ( y ) ( - + 1) GO@, y ) b Y . C’

It follows from the integral equation v + Pov = 0 on C’ and the properties a,,) - eo) of Go(x, y), verified in section 3, that

V E P(D) n c(D) , A V = O i n D , V = O o n C = C l + C ’ , V = O ( I ) , V V = ~ ( r - ’ ) asr=Ixl+oc).

(4.7)

These properties imply, by a well-known argument due to R. Leis [4] (compare also [2], Theorem 3.27), that V E C’ (fi). Hence Green’s formula can be applied to V in D and yields V = 0 in D as in the proof of Theorem 2.1. Consider the


interior D[:= Di2 u . . . u Dim of C'. It follows from Q) and classical theorems on single and double layer potentials that Y E C'@J and that V satisfies the jump relations

a a (4.8) - Vj - - V, = 2 v ,

a n a n

Since V = 0 in D, (4.8) implies

Vj - V, = -2v o n C ' .

that

Vi = -6 o n C ' a a n -

and hence, by Green's formula,

so that 6 = 0 and v = (Ye - 6) /2 = 0 on C'. This completes the proof of Lemma 4.1.

By Lemma 4.1 and (4.4), ( I + Px)-' exists in a neighborhood of x = 0. Furthermore, it follows by a local Neumann expansion that

Let Li and LA be the integral operators introduced in (1.2) and (1.4), respectively, with C, replacing C. By applying Theorem 2.1 to C, , we conclude that ( I + Li)-' exists in a neighborhood of x = 0. Since1lL.i - LA11 = O(l/Ilnxl) by (3.21), we obtain in analogy to (4.9)

For sufficiently small 1x1, the solution Ui of the Dirichlet problem for A U + x2 U = 0 in the exterior D, of C, is given by

with px = ( I + L.;)-' y x . Since

(4.12) l lyx - yell:= suprIy,(x) - YO(X)~-+ 0, X E c

it follows from (4.10) that Ui converges to

+ 1 h,, forxED, (4.13) UA(x) := - j po(y) --In- ) * =1 (aiy 1 X - Y )

1

148 P. Werner

as x -+ 0 with (4.14) po := ( I + LA)-'yo. By (4.10) and (4.12), we have

and hence

uniformly in every compact subset of D, . In particular, (4.16) holds uniformly on C' = C, + . . . + C,,,. The estimates (4.9), (4.12) and (4.16) imply that the solution v, = (I + P,)-' (y, - U!J of the integral equation (4.3) converges to

as x -+ 0 and that (4.17) UO := ( I + Po)-'(yo - Uh)

By (1.QY (4.16), (4.18). (3.33) and (3.34). the solution V, of the Dirichlet problem (1.6) converges to

and satisfies the estimate

(4.20) U,(X) - U ~ ( X ) = 0

uniformly in every compact subset of D. Uh is the solution of the exterior Dirich- let problem for A U = 0 in D, with boundary data yo on C, since po + LLpo = yo on C, by (4.14). Hence (4.19) implies Uo = UA = yo on CI . Since vo + Povo = yo - Uh on C' by (4.17), the boundary condition Uo = yo holds also on C' = C - C, . This, in connection with the properties %) - eo) of Go(x, y ) , shows that Uo is the solution of the boundary value problem (1.10). Thus we have proved:

Theorem 4.1 Assume that D is the exterior of m twice continuously differentiable closed curves C, , . . . , C, and that yx k a family of continuous functions, defined on C = C, + . . . + C,,,, with yx yo asx -, 0 uniformIy on C. Then the solution U, of the exterior Dirichletproblem (1.6) for the reduced wave equation A Ux + x2 U, = 0 converges to the solution Uo of the exterior Dirichlet problem (1 .lo) for A Uo = 0 as x -+ 0. Furthermore, the estimate (4.20) holds uniformly in every compact subset of D.


The estimate (4.20) for U, - Uo cannot be improved, as the example C = { x : l x l = l ) , yx = 1 shows. In this case we have by (1.3)

The logarithmic order of the remainder stands in noticeable contrast to the results on low frequency approximations for space dimensions n 2 3 . It is easy to deducefrom[7] or[1] that U, - UO = O(Ix1 + ) I y x - y o I ) ) a s x + Oi fn Z 3 .

Now we apply the methods, developed above, to the extension of the results in [8] to the case rn > 1. Let Ux = U,[f] be the solution of the exterior Dirichlet problem

[(A + x 2 ) U , = -f i n D , U x = 0 o n C ,

for a given functionfe Ch(D). We extend f to the exterior D1 of Cl , by setting f = 0 in D, - D, and consider the volume potential (4.23) T x ( x ) := S f (y)G,(x, Y ) ~ Y

where G,(x, y ) denotes as above Green’s function for A U + x2U = 0 in the exterior D, of C, , The properties a) - e) of G,(x, y) , stated in section 3, imply that

Dl

( A T , + x2T, = -f i n D , , .Tx = 0 onC, ,

L

By (3.33) we obtain

B Inx

(4.25) T, = To + 2 + 0

uniformly in every compact subset of D1, where

By a,) - eo) we have

(4.27)

I t follows from (4.22) and (4.24) that

is the solution of the exterior Dirichlet problem

A T o = -f i nD1 , To=O o n C , , I T o = O ( l ) , V T O = O ( ~ - ~ ) % r = l x l - + m .

(4.28) V, := U, - T,

((A + x 2 ) V, = o i n D , V,=O o n C l , V x = -Tx o n C ’ ,

150 ~ . w c r n c r

Since T, -+ To uniformly on C', Theorem 4.1 implies that V, converges to the solution Vo of the boundary value problem

(4.30)

Thus we have shown that U, = V, + T, converges to the solution Uo = Vo + To of

(4.31)

uniformly in every compact subset of D.

term of the remainder U, - Uo. According to (1.7) - (1.9) and (4.2), we have

A V o = O i n D , V o = O o n C , , Vo= -To onC' , V, = 0 ( 1 ) , v V, = o(r -2 ) as f = 1x1 -* 00 .

A U o = -f i n D , Uo=O o n C , I u , = o ( I ) , V U , = O ( ~ - ' ) asr=Ixl-+cO

In view of the applications to (81, it is necessary to specify the leading

with (4.33) V, = -(I + P,)-' T,. Note that (4.32) and (4.33) hold also for x = 0. By (3.33), (3.34) and (4.2), we obtain

1 lnx

(4.34) P, - Po = - Q + R,

with

(4.35) (Qv) (x ) = 2 5 v(y) A(x, y)dsy forxtz C' c'

andllR,II = O(l/llnx12) a s x 4 0. The first two terms of the Neumann expansion of (I + = [ ( I + Po) + (P, - P,,)]-' yield by (4.25) and (4.34)

V, = - ( I + P,)-' T, 1

Inx = -(I+Po)- 'T,+-(I+Po)- 'Q(Z+Po)-'T,+ 0

0 + o - = Y o + - In x

uniformly on C' with

(,I&) asx-,

u : = - ( I + PO)-'B, + ( I+Po)- 'Q(I+Po)- 'To.


By inserting the last estimate into (4.32) and observing (3.33) and (3.34), it follows that

uniformly in every compact subset of D, where B2 is given by

Combining (4.25) and (4.36), we obtain

(4.37) u, = uo + - In x

with B := B, + B2. Note that B, , B2 and B are continuous in D. Thus we have proved:

Theorem 4.2 Assume that D is the exterior of rn twice continuously differentiable closed curves C, , . . . , C, and that f E CA(D). Then the solution U, of (4.22) converges to the solution Uo of (4.31) as x + 0. Furthermore, the estimate (4.37) holds uniformly in every compact subset of D, where B is continuous in D.

Theorem 4.2 extends [8], Lemma 4.2 to the case m > 1. The argument, used in the last part of section 4 in 181, yields:

Corollary to Theorem 4.2 Satz 4.1 and Satz 4.2 in [8] on the asymptotic behavior of the solutions u (x, t ) of the exterior Dirichlet problems for the wave equation and the heat equation as t 4 CD remain valid if C consists of a finite number of closed curves C, , . . . , C,,, .

Now we want to extend the results on the low frequency behavior of Green’s function in section 3 to the case m > 1. In the following we denote Green’s function for the exterior D1 of C1 by GA(x, y ) and Green’s function for the exterior D of C = C, + . + C, by G,(x, y ) . Near x = 0, G,(x, y ) is given by

(x, y E D) with

(4.39) v X ( . , y ) : = - ( I + P,)-’G;(.,y). In fact: By (4.39), v, satisfies the integral equation

(4.40) v,(. , U) + P,v,(. , y ) = - GA(. , y) on C’ ,

152 P. Werner

so that G,(x, y ) -, 0 as x -. xo E C’ by (4.38) and the jump relation. Hence the properties a) - d) for G,(x, y ) follow from the corresponding properties for GL(x, y) , which have been verified in section 3. Note that the estimates (3.33) and (3.34) hold for GL(x, y ) uniformly in every compact subset of D1. Thus we conclude from (4.38), (4.39) and (4.9):

Theorem 4.3 Assume that D is the exterior of m twice continuously differentiable closed curves CI , . . . , C,,, . Then Green ’s function G,(x, y ) of the exterior Dirichlet problem for A U + x2 U = 0 converges to the corresponding Green’s function Go(x, y ) for A U = 0 as x -+ 0. G,(x, y ) and Go(x, y ) are uniquely determined by properties a) - d) and %) - do), respectively, stated in Section 3 (with C = Cl + . . + C,,,). Furthermore, the estimate

holds uniformly in Kl x Kz for every pair of disjoint compact subsets K, , K2 of D.

We conclude this paper with some remarks on Green’s function Gi (x , y ) of the exterior Neumann problem for A U + xz U = 0 in the exterior D of C = C1 + - + C,,, . In analogy to (3.3) - (3.6), G;(x, y ) is given by

1 (4.42) G:(x, y ) = - -Hi”(x[x - y l ) + M;(x, y ) 4

with

and

. a -H&l’(xlx - Y l ) for x E C a n d y E D . 1 (4.46) Y ~ ( X , y ) := --- 4 an,

Set

fo rxE C 1 a 1 71 c an, [ x - z [

(4.47) (L&) (x ) : = - j p (t) -In - ds* and

(4.48) y6(x, y ) := -- -In- 2n an, [ x - y [ i a 1

for X E C , y E D


and note that

(4.49) 1 1 ~ : - till = o

and

(4.50) y:(x, u) - r&x, Y ) = 0

uniformly in C x K for every compact subset K of D. It is a well-known result of classical potential theory that ( - I + &)-' exists for every m 2 1. Hence also ( - I + t i)- ' exists for sufficiently small I x 1, and

(4.51) //(-I + L;)-l - ( - I + Ld)-'II = 0 ( IxI2ln- ,iI) a s x - 0 .

It follows from (4.44) - (4.51) that pX = (-I + L;)-' y; converges to

a s x + O

(4-52) P O ( . , Y ) := ( -1 + L~)-'Y~(.,Y) as x -, 0 and that

(4.53) px(z, u) = P O ( Z , u) + 0

uniformly in C x K for every compact subset K of D. By (4.43), (4.53) and (1.3), we have

i (4.54)

l a s x -. 0

uniformly in K x K for every compact subset K of D. Set

The jump relation implies that

f o r x E C a n d y e D .

Since ( - I + Li)po( . , y ) = y d ( . , y ) by (4.52), we have

P. Werner 154

(4.56) i a 1 (xty) = yA(x,y) = -- -In- 2n an, Ix -y l

f o r x E C a n d y e D ,

and hence

(4.57) po(xJ y ) = - Ui(x, y ) f o r x e C a n d y € D

with

a 8%

forxE DiandyE D .

Since A,U(x, y ) = 0 for x E Dj and y E D, it follows from (4.57) that

(4*59) jpo(xJ y)dsX = j A.XU(x, yldu = c Di

for every y E D. By (4.54), (4.55) and (4.59), we obtain

(4.60) M;(x, y ) = Mh(xJ y ) + 0

uniformly in K x K for every compact subset of D. Thus, in contrast to the Dirichlet case, the compensating term M;(xJ y ) of Green’s function G;(x, y ) for the exterior Neumann problem converges as x -+ 0. Consequently, G;(xJ y ) has no limit as x -, 0. By (4.42)’ (4.60) and (1.3), we have

1 1 (4.61) Gi(x, y ) = -In- + G,$(x, y ) + 0

2R X

uniformly in Kl x K2 for every pair of disjoint compact subsets Kl, K2 of D, where Gd(x, y ) is given by

+ In2 - y + - (4.62) G~(x, y ) = - In - 1

Gd(x, y ) has the following properties: r( . 1 1

a61 Gd(x, u) - - In . has continuous derivatives of arbitrary order in D x D; 272 Ix -u l bi) A$h(x, y ) = 0 if x, y E D and x * y ;

1 1 ci) Gi(x, y ) = -In-!- + - ( In2 - y + - y ) + O(+),

271 1x1 2n

Low Frequency Asyrnptotics for the Reduced Wave Equation 155

l x V,G,j(x, y ) = -- - + O ( r - 2 ) asr = Ixl+ 03 2n 1x12

uniformly with respect toy in every compact subset of D; db) for fiied y E D, Gd(. , y ) can be extended to a continuously differentiable function on 6, satisfying the boundary condition

a 8%

(4.63) -G(x,y) = 0 f o r x E C .

Compare the proof of property a) in section 3 for the verification of a(,); bb) follows immediately from (4.62) and (4.55). Formulas (4.55) and (4.59) imply Md(x, y ) = O(r - ' ) and V'd(x, y ) = O ( r - 2 ) as r = 1x1 -+ 03, so that cb) holds by (4.62). The boundary condition (4.63) follows from (4.62) and (4.56). Note that Gb(x, y ) is uniquely determined by G)-db). In fact: Assume that also Gd'(x, y ) satisfies properties ab)-d$ and consider, for fixed y E D, U(x) := G;(x,y) - Gi'(x,y).Thenwe have A U = O i n D , ( a / a n ) U = Oon Cand I/ = O(r- ' ) , V U = O ( r - 2 ) as r = 1 X I + 03, so that U = 0 in D by Green's formula. Thus we obtain, in agreement with [6], Theorem 3, case 11:

Theorem 4.4 The asymptotic behavior of Green's function GL(x, y ) of the exterior Neumann problem for the reduced wave equation A U + x2 I/ = 0 as x -. 0 is given by (4.61). In particular, we have 1 GL(x, y ) I m as x + 0 for all x, y E D with x + y . The second term Gd(x, y ) in the asymptotic expansion (4.61) is uniquely determined by the properties G) - db), listed above.

The comparison of Theorem 4.3 and Theorem 4.4 exhibits significant differences, with regard to the low frequency behavior of the solutions, between the Dirichlet and the Neumann problem for the reduced wave equation in two- dimensional exterior domains. Relations to the time-dependent theory, in particular to the different asymptotic behavior of the solutions of the corresponding initial and boundary value problems for the two-dimensional wave equation as t -+ 03, are studied in [8].

References

Brakhage , H.; Werner , P.: Ober das Dirichletsche AuRenraumproblem filrdic Helrnholtzsche Schwingungsgleichung. Arch. d. Math. 16 (1965) 325- 329 C o l t o n , D.; Kress , R.: Integral Equation Methods in Scattering Theory. New York: John Wiley 1983 Kr ess , R.: On the limiting behavior of solutions to boundary integral equations associated with time harmonic wave equations for small frequencies. Math. Meth. in the Appl. Sci. 1 (1979) 89-100 Leis , R.: uber die Randwertaufgaben des Aunenraumcs zur Helmholtzschcn Schwingungs- gleichung. Arch. Rational Mech. Anal. 9 (1962) 29-44 MacCamy, R. C.: Low frequency acoustic oscillations. Quart. Appl. Math. 23 (1965) 247-256 Murave i , L. A.: Analytic continuation with respect to a parameter of the Green's functions of exterior boundary value problems for the two-dimensional Helmholtz quation 111. Mat. Sbornik 105 (147) (1978) 63-108. Engl. Obersctzung: Math. USSR Sbornik 34 (1978) 55-98

156 ~.Werner

(71 Werner, P.: Randwertprobleme der mathematischen Akustik. Arch. Rational Mech. Anal. 10

[8] Werner, P.: Zur Asymptotik der Wellengleichung und der WBrmeleitungsgleichung in zwei-

[9] Ramm, A. G.: The behavior of the solutions to exterior boundary value problems at low fre-

(1962) 29-66

dimensionalen AuRenrBumen. Math. Meth. in the Appl. Sci. 7 (1985) 170-201

quencies, 2 preprints (1985)

Prof. Dr. Peter Werner Mathematischa Institut A Universitit Stuttgart Pfaffenwaldring 57 D-7000 Stuttgart 80 (Received October 29, 1984)

low frequency asymptotics for the reduced wave equation in two-dimensional exterior spaces

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