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Existence and Uniqueness of Scattering Solutions in Non-smooth Domains
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
201, 329]338 Ž1996.
0258
Existence and Uniqueness of Scattering Solutions in Non-smooth Domains A. G. Ramm* Department of Mathematics, Kansas State Uni¨ ersity, Manhattan, Kansas 66506-2602
and A. Ruiz† Department de Matematicas, Uni¨ ersidad Autonoma de Madrid, 28049 Madrid, Spain ´ ´ Submitted by William F. Ames Received October 3, 1995
A short and self-contained proof of the existence of the scattering solution in exterior domains is presented for some class of second order elliptic equations. The method does not use the integral equation; it is based on Fredholm theory and the limiting absorption principle for solutions in the whole space. It covers domains with Lipschitz boundaries, domains satisfying a cone condition, and those with the so-called local compactness property. Q 1996 Academic Press, Inc.
1. INTRODUCTION Consider the scattering problem Lu [ lu q K 2 u s 0 Gu s 0 u s u0 q ¨ ,
in D9
Ž 1.
on S,
Ž 2.
u 0 [ exp Ž iK a ? x . , a g S ny 1 ,
Ž 3.
* E-mail address: [email protected]. † Second author is supported by Spanish DGYCT PB94-0192. 329 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
330
RAMM AND RUIZ
where ¨ satisfies the radiation condition lim
H
rª` < x
< ¨ r y iK¨ < 2 ds s 0,
Ž 4.
D ; R n, n ) 2 is a bounded domain not necessarily connected, D9 [ R n _ D, S s D, Gu is a boundary condition which is assumed to be one of Gu s u
Ž Dirichlet BC. ,
Ž 2D.
Gu s u N
Ž Neumann BC. ,
Ž 2N.
where u N is the conormal derivative with respect to the principal part ˜ l of the operator l, ˜ lu [ i Ž a i j Ž x . j u., i.e., u N [ a i j u j Ni , u j s ur x j Žhere and below summation is understood over the repeated indices., Gu s u N q s Ž s . u
Ž Robin BC. .
Ž 2R.
Here N is the unit exterior normal to S, s Ž s . is a continuous function on S, s g S, K ) 0 is a constant, and lu [ i Ž a i j Ž x . j u . y q Ž x . u
Ž 5.
is an elliptic formally symmetric, with real-valued coefficients, differential operator in R n ; i.e., for every t g C n there exist possitive constants C1 , C2 such that C1 t j t j F a i j Ž x . t i t j F C 2 t j t j ,
a i j s a ji s a i j .
Ž 6.
if < x < ) a, a i j Lipschitz continuous,
Ž 7.
We assume that ai j s di j q s q,
q is an L` y function compactly supported in the ball < x < - a, Ž 8. 0 F s Ž s. g CŽ S . .
Ž 9.
Conditions Ž7. and Ž8. are sufficient for the unique continuation property ŽLemma 2 below. to hold. At the end of the paper in Remark 1 we discuss possible generalizations. In the sequel we denote by Br the ball of radius r centered at the origin, Sr [ Br and Dr [ D9 l Br , r ) a, and assume that S ; Ba . Lw.,.x [ a i j Ž.. i Ž.. j denotes the bilinear form associated with the principal part of L. The classical formulation of the problem Ž1. ] Ž4. is clear if S is sufficiently smooth. If S is not smooth Žrough domain. then it is necessary to define the meaning of Ž2..
UNIQUENESS THEOREMS
331
The following assumption ŽA. concerning the class of domains D will be crucial: ŽA1. For r ) a the embedding i : H 1 Ž Dr . ª L2 Ž Dr . is compact. ŽA2. The trace operator r : H 1 Ž Dr . ª L2 Ž S . exists and is compact. These assumptions restrict the smoothness of D implicitly. They are rather weak and are satisfied in most of the practically interesting cases: Lipschitz domains and domains with cone property are admissible, as we explain below. Assumption ŽA2. is needed only for the Robin condition; we do not use it for the Neumann condition, while for the Dirichlet condition no assumptions on D are needed, except boundedness of D. Assumptions ŽA. hold, for instance, if D is a Lipschitz domain D g C 0, 1 , that is, a domain whose boundary is locally the graph of a Lipschitz function, and the Lipschitz constant does not depend on the local patch of the boundary. Also the following cone property for D suffices for ŽA. to hold: For any p g D there exists a cone K Žwith the vertex at p . contained in D together with its closure, K [ x : < x9 < 2 F bx n2 , 0 - x n - a; b ) 0, x9 s Ž x 1 , . . . , x ny1 . 4 . The class of domains having the cone property is larger than the class of Lipschitz domains as defined above. Assumption ŽA. holds also if D g EV21 , the class of domains for which there exists a bounded extension operator E : H 1Ž D. ª H 1ŽR n. ,
Eu Ž p . s u Ž p . if p g D.
This class also contains the Lipschitz domains. If we deal with boundary condition Ž2N. then we only consider assumption ŽA1. on S. For the Dirichlet problem we can drop the whole assumption ŽA. and assume just that D is a compact domain. In w2, p. 243x a necessary and sufficient condition for ŽA1. to hold is given. In w1x the boundary-value problems for second order elliptic equations in bounded Lipschitz domains are studied. We prove in the present work, that for the class of compact domains satisfying condition ŽA. appropriate to the boundary condition, the solution to Ž1. ] Ž4., as defined above, exists and is unique. In the proof we will
332
RAMM AND RUIZ
use two well known results: LEMMA 1 Ž Rellich’s Type Lemma, see w3, p. 25x.. Let u be a solution of Ž D q K 2 . u s 0 in < x < ) R, such that H< x
2. UNIQUENESS Let us start with the definition of the solution for general domains. DEFINITION 1. We say that u solves the scattering problem if, for every r ) a, Ža. u g H , Žb. for all test functions w g H 1 Ž D9., vanishing near infinity, the following integral identity holds,
HD9 ž K Žc.
2
uw y quw y a i j u j wi dx q
/
HS s uwds s 0,
Ž 10 .
u satisfies Ž3. and Ž4..
If Neumann Ž2N. or Robin condition Ž2R. are imposed then H s H 1 Ž Dr .. For the Neumann boundary condition the boundary integral in Ž10. is dropped. If the Dirichlet condition Ž2D. is imposed then H s H˚ 1 Ž Dr . Ž H˚ 1 Ž Dr . is the space of functions in H 1 Ž Dr . vanishing on S ., the boundary integral in Ž10. is dropped, and the test function w has to be taken in H˚ 1 Ž D9..
333
UNIQUENESS THEOREMS
Condition Ž4. makes sense: from Ž7., Ž8., and the regularity results for the weak solutions of Ž1., it follows that ¨ is smooth in DaX . Clearly if D is a Lipschitz domain, then Ž10. implies Eq. Ž1. and the boundary condition Ž2.. If s s 0 then the last integral in Ž10. is absent. The easier case of the Dirichlet condition is left to the reader as an exercise. In this case, the last integral in Ž10. vanishes and w runs through the subspace of H 1 Ž DrX . of functions vanishing on S in the sense of the embedding theorem. Let us state the main result of this section. THEOREM 1. If D is compact and assumptions Ž6. ] Ž9. hold, then the solution to the scattering problem is unique. Proof. We first pass from the homogeneous equation Ž1. to the equivalent non-homogeneous one in order to eliminate u 0 from the asymptotic condition at infinity Ž3. and Ž4.. This passage is standard. Fix r 0 ) a, take h g C`ŽR., such that h9Ž r . G 0, hŽ r . s 1 if r ) r 0 q 1, hŽ r . s 0 if r - r 0 , and define W Ž x . s ¨ Ž x . q Ž1 y hŽ< x <.. u 0 Ž x .. Inserting u s W q hu 0 in Ž10., using Ž8., Ž9., and Green’s formula, we obtain
HD9 ž K
2
Ww y qWw y a i jWj wi q fw dx q
/
HS s Ww s 0,
Ž 11 .
where f [ Ž D q K 2 .Ž hu 0 . is a C` function supported in r 0 - < x < - r 0 q 1. Since W Ž x . s ¨ Ž x . if < x < ) r 0 q 1, W satisfied the radiation condition lim
H
rª` < x
< Wr y iKW < 2 ds s 0.
Ž 12 .
To prove the uniqueness of the solution it is enough to check that from Ž11. with f s 0, and Ž12., it follows that W s 0. To do this, we use the standard strategy w3x. From Ž12. we obtain lim rª0
½H Ž Sr
K 2 < W < 2 q < Wr < 2 . ds q iK
HS Ž WW y WW . ds r
r
r
5
s 0.
Ž 13 .
If we prove
HS Ž WW y WW . ds s 0, r
r
Ž 14 .
r
for r ) a, then Ž13. and Lemma 1 imply that W s 0 in R n _ Ba , and W s 0 in D9 from Lemma 2. To derive Ž14., take in Ž11., with f s 0, the test function w s WhŽŽ< x < y r 0 .re . [ Whe , r 0 ) a, where h g C`ŽR., h G 0, hŽ r . s 0 if r - 1r2 and hŽ r . s 1 if r ) 3r2.
334
RAMM AND RUIZ
Subtract from expression Ž11. its complex conjugate. The real-valuedness of q, s , and K, and the assumptions on a i j , imply
HD9 ž W W y W W / Ž h . j
j
e
j dx
s 0.
We have Ž he . j s h9ŽŽ< x < y r 0 .re .Ž x jrŽ e < x <.., Ž1re . h9ŽŽ< x < y r 0 .re . ª d Ž< x < y r 0 ., and W g C`Ž< x < ) a.. Therefore we can take e ª 0 and get Ž14.. Note that in this argument assumption ŽA. is not used. Theorem 1 is proved. The following lemma will be used in Section 3. LEMMA 3. If W is a solution of the equation LW s 0 in R n, which satisfies radiation condition Ž12., then W s 0. Proof. As in the proof of Theorem 1, if suffices to prove Ž14.. We have W LW y W LW s 0. Integrate over Br and use Green’s formula to get 0s
HB Ž Lw W , W x y Lw W , W x . dx q HS Ž WW y WW . ds. r
r
r
r
Assumptions Ž6. and Ž8. imply that the first integral vanishes, we get Ž14., and Lemma 1 implies W s 0.
3. EXISTENCE THEOREM 2. If assumptions Ž A. and Ž6. ] Ž9. hold, then there exists a solution to the scattering problem and this solution is unique. As in Theorem 1, it is sufficient to prove the existence of a function W g H such that Ž11. and Ž12. hold with f supported in the annulus r 0 - < x < - r 0 q 1, r 0 ) a. We give the argument for the Neumann condition Ž s s 0., the case of the Robin condition is treated similarly. The idea of the proof is to reduce the scattering problem to a Fredholm-type equation without using integral equations, and to derive the existence of its solution from the uniqueness of the solution, which is a consequence of Theorem 1. We first prove an auxiliary result stated in Proposition 1, then describe the above reduction, and then complete the proof of Theorem 2. Let us pass to the auxiliary result. Consider the problem LV s h in R n ,
V satisfies the radiation condition Ž 4 . .
Ž 15 .
UNIQUENESS THEOREMS
335
Denote H 0 [ L20 Ž DR ., R ) r 0 q 1, the set of L2 Ž DR . functions vanishing near SR , and assume that supp h ; DR . PROPOSITION 1. Gi¨ en an arbitrary h g H 0 there exists a unique solution V of Ž15. such that V g Hl2o c and 5 V 5 b - `, b ) 1, where 5 V 5 2b [
HR
Ž 1 q < x <.
b
dx.
Proof of Proposition 1. From Ž6. ] Ž8. it follows that L is a symmetric, semibounded from below, operator in L2 with domain H02 , the set of H 2 ŽR n . functions vanishing near infinity. We can take its Friedrichs’ extension, also called L, defined on the dense subset of the domain of the quadratic form associated to L q C ) 0, which is H 1. From the self adjointness of L it follows that for e ) 0 the function Ve [ Ž L q i e .y1 h is well defined. The following lemma yields the conclusions of Proposition 1. LEMMA 4. We ha¨ e Ve ª V
as e ª 0 in Hl2o c and in 5 5 b ,
Ž 16 .
and V sol¨ es Ž 15 . .
Ž 17 .
We prove Lemma 4 in two steps. Step 1. Under the assumption sup 5 Ve 5 b - `,
Ž 18 .
0- e - e 0
the assertions Ž16. and Ž17. hold. Step 2. Under the hypothesis of Proposition 1, inequality Ž18. holds. If Ž18. holds, then for any ball BR , 5 Ve 5 L2 Ž B R . - C Ž R ., and therefore Ve converges weakly in L2 Ž BR . to some function V. We have LVe s yi e Ve q h, 5 LVe 5 L2 Ž B R . - C Ž R, h., and from the interior elliptic estimates it follows that 5 Ve 5 H 2 Ž BXR . - C Ž R, R9, h.; R9 - R. Therefore, for any R - a, 5 Ve 5 L2 Ž B R . - C Ž R ., Ve ª V in H s Ž BR ., s - 2. Using again Ž15., we conclude that LVe converges in L2 Ž BR ., and from the interior elliptic estimates we know that 5 Ve y Ve 9 5 H 2 Ž BXR . F C Ž 5 LVe y LVe 9 5 L2 Ž B R . q 5 Ve y Ve 9 5 L2 Ž B R . . .
336
RAMM AND RUIZ
It follows that Ve ª V strongly in Hl2o c . We can pass to the limit in the equation and get LV s h. We now prove convergence in the norm 5 5 b and verify the radiation condition for V. In BRX , R ) a, Ve satisfies the equation Ž D q K 2 q i e .Ve s 0. Therefore, Green’s formula yields Ve Ž x . s
HS
x g BRX ,
Ve Ž s . Ž ge Ž x y s . . r y Ž Ve . r Ž s . ge Ž x y s . ds, R
Ž 19 . where ge is the fundamental solution ge Ž x . [
Ž ie q K 2 . < x<
Ž ny2 .r2
vn
Ž ny2.r2
2 HŽŽ1. ny2.r2 Ž i e q K .
ž
1r2
< x< ,
/
and we take I Ž i e q K 2 .1r2 ) 0. From Ž19. it follows that < Ve Ž x . < F
c < x<
Ž ny1.r2
,
if < x < ) R 0 , 0 - e - e 0 .
Ž 20 .
Since Ve ª V strongly in Hl2o c , we can pass to the limit in Ž19. and get VŽ x. s
HS
V Ž s . Ž g 0 Ž x y s . . r y Ž V . r Ž s . g 0 Ž x y s . ds,
x g BRX .
R
From this representation of V it follows that V satisfies the radiation condition. Finally, from estimate Ž20. and the pointwise convergence Ve ª V it follows that 5 Ve y V 5 b ª 0 and Step 1 is completed. Let us pass to Step 2. Arguing by contradiction, assume the existence of sequences e k ª 0, Vk [ Ve k, such that 5 Vk 5 b ª `. Since Ve s Ž L q i e .y1 h and 5 Ve 5 b F Ž1re .5 h 5 L2 , we can define Wk [ Vkr5 Vk 5 b . Then
Ž L q i e . Wk s
h 5 Vk 5 b
[ hk
and
5 h k 5 L2 Ž B R . ª 0.
Ž 21 .
Since 5 Wk 5 b s 1, under condition Ž21. we can repeat the arguments in Step 1 and prove that Wk ª W in Hl2o c , W satisfies the radiation condition, 5 Wk y W 5 b ª 0, and LW s 0. From the uniqueness Lemma 3 it follows that W s 0. This contradicts the relations 5 Wk y W 5 b ª 0 and 5 Wk 5 b s 1. Proposition 1 is proved.
337
UNIQUENESS THEOREMS
Let us now describe the reduction of the scattering problem in the form Ž11. ] Ž12. to a Fredholm-type equation Žw3, p. 36x.. Set W [ V y h Z,
Ž 22 .
where V solves Ž15. and h is a C` function which equals 1 near S and 0 outside DR . In particular, h s 0 on SR . The function W solves problem Ž11. ] Ž12. if Z solves the problem yf s h y L Ž h Z . in DR ,
GŽ Z. s GŽ V . ,
Z s 0 on SR , Ž 23 .
where we have used the strong formulation of the problem for convenience of the reader and with the understanding that Ž23. is understood in the weak sense, similarly to Ž11.. Note that near infinity W, defined in Ž22., equals V and, by the definition of V, satisfies the radiation condition Ž12.. There are many Z which solve Ž23. Žsince h in Ž23. is arbitrary .. Let us fix a unique Z as the solution to the problem LZ s iZ in DR ,
G Ž Z . s G Ž V . on S,
Z s 0 on SR ,
Ž 24 .
where again the weak formulation via the integral identity is understood. Clearly the solution to Ž24. is a linear operator on h. Define Bh [ LŽh Z .. Then Eq. Ž23. can be written as h y Bh s yf .
Ž 25 .
The operator B in Ž25. is compact in H 0 . Indeed, LŽh Z . s h LZ q QZ, where QZ [ LŽh Z . y h LZ contains not higher than the first derivatives of Z and LZ s iZ by Ž24.. The map h ª V ª Z ª QZ is the map H 0 ª H 1, as follows from the known estimates for the solutions of second order elliptic equations in bounded domains Žrecall that QZ vanishes near non-smooth boundary S because h s 1 near S .. By assumption ŽA1., the embedding H 0 ª H 1 is compact, so B is compact in H 0 . If the Robin boundary condition is used, then we use assumption ŽA2. also. Suppose that h solves Ž25., V solves Ž15., and W is defined by Ž22.. Then W solves Ž11. ] Ž12., as we checked above. Therefore, the scattering problem in the form Ž11. ] Ž12. is reduced to Eq. Ž25. with compact operator B in H 0 . The proof of Theorem 2 will be completed as soon as we show that the homogeneous version of Eq. Ž25. has only the trivial solution. Let us show this and thus complete the proof of Theorem 2. Assume that h solves Ž25. with f s 0. Then W, defined by Ž22., solves Ž11. ] Ž12. with f s 0. By Theorem 1, we get W s 0 in D9. Thus, V s h Z in DR , and V s Z on S. This and the first boundary condition Ž24. imply that V and Z have the same Cauchy data on S. Therefore, the function Z,
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RAMM AND RUIZ
extended into D so that Z s V in D, solves the problem LZ s iZ in DR ,
LZ s 0 in D,
Z s 0 on S R ,
Ž 26 .
where the equation LZ s 0 in D follows from Ž15. and the fact that h s 0 in D. Since L is symmetric, it follows from Ž26. that Z s 0 in BR . Therefore V s 0 in DR , and h s LV s 0 in DR . Theorem 2 is proved. Remark 1. We can relax condition Ž8.: allow for q g L lpo c , p ) nr2, and allow q s oŽ< x
REFERENCES 1. C. Kenig, ‘‘Harmonic Analysis Techniques for Second Order Elliptic Boundary Problems,’’ Amer. Math. Soc., Providence, Rhode Island, 1994. 2. V. Mazja, ‘‘Sobolev Spaces,’’ Springer-Verlag, New York, 1985. 3. A. G. Ramm, ‘‘Scattering by Obstacles,’’ Reidel, Dordrecht, 1986. 4. A. G. Ramm, Uniqueness theorems for inverse obstacle scattering problems in Lipschitz domains, Appl. Anal., 59, Ž1995., 377]383. 5. C. Wilcox, Scattering theory for the D’ Alembert equation in exterior domains, in ‘‘Lecture Notes in Math.,’’ Vol. 442, Springer-Verlag, New York, 1975.
201, 329]338 Ž1996.
0258
Existence and Uniqueness of Scattering Solutions in Non-smooth Domains A. G. Ramm* Department of Mathematics, Kansas State Uni¨ ersity, Manhattan, Kansas 66506-2602
and A. Ruiz† Department de Matematicas, Uni¨ ersidad Autonoma de Madrid, 28049 Madrid, Spain ´ ´ Submitted by William F. Ames Received October 3, 1995
A short and self-contained proof of the existence of the scattering solution in exterior domains is presented for some class of second order elliptic equations. The method does not use the integral equation; it is based on Fredholm theory and the limiting absorption principle for solutions in the whole space. It covers domains with Lipschitz boundaries, domains satisfying a cone condition, and those with the so-called local compactness property. Q 1996 Academic Press, Inc.
1. INTRODUCTION Consider the scattering problem Lu [ lu q K 2 u s 0 Gu s 0 u s u0 q ¨ ,
in D9
Ž 1.
on S,
Ž 2.
u 0 [ exp Ž iK a ? x . , a g S ny 1 ,
Ž 3.
* E-mail address: [email protected]. † Second author is supported by Spanish DGYCT PB94-0192. 329 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
330
RAMM AND RUIZ
where ¨ satisfies the radiation condition lim
H
rª` < x
< ¨ r y iK¨ < 2 ds s 0,
Ž 4.
D ; R n, n ) 2 is a bounded domain not necessarily connected, D9 [ R n _ D, S s D, Gu is a boundary condition which is assumed to be one of Gu s u
Ž Dirichlet BC. ,
Ž 2D.
Gu s u N
Ž Neumann BC. ,
Ž 2N.
where u N is the conormal derivative with respect to the principal part ˜ l of the operator l, ˜ lu [ i Ž a i j Ž x . j u., i.e., u N [ a i j u j Ni , u j s ur x j Žhere and below summation is understood over the repeated indices., Gu s u N q s Ž s . u
Ž Robin BC. .
Ž 2R.
Here N is the unit exterior normal to S, s Ž s . is a continuous function on S, s g S, K ) 0 is a constant, and lu [ i Ž a i j Ž x . j u . y q Ž x . u
Ž 5.
is an elliptic formally symmetric, with real-valued coefficients, differential operator in R n ; i.e., for every t g C n there exist possitive constants C1 , C2 such that C1 t j t j F a i j Ž x . t i t j F C 2 t j t j ,
a i j s a ji s a i j .
Ž 6.
if < x < ) a, a i j Lipschitz continuous,
Ž 7.
We assume that ai j s di j q s q,
q is an L` y function compactly supported in the ball < x < - a, Ž 8. 0 F s Ž s. g CŽ S . .
Ž 9.
Conditions Ž7. and Ž8. are sufficient for the unique continuation property ŽLemma 2 below. to hold. At the end of the paper in Remark 1 we discuss possible generalizations. In the sequel we denote by Br the ball of radius r centered at the origin, Sr [ Br and Dr [ D9 l Br , r ) a, and assume that S ; Ba . Lw.,.x [ a i j Ž.. i Ž.. j denotes the bilinear form associated with the principal part of L. The classical formulation of the problem Ž1. ] Ž4. is clear if S is sufficiently smooth. If S is not smooth Žrough domain. then it is necessary to define the meaning of Ž2..
UNIQUENESS THEOREMS
331
The following assumption ŽA. concerning the class of domains D will be crucial: ŽA1. For r ) a the embedding i : H 1 Ž Dr . ª L2 Ž Dr . is compact. ŽA2. The trace operator r : H 1 Ž Dr . ª L2 Ž S . exists and is compact. These assumptions restrict the smoothness of D implicitly. They are rather weak and are satisfied in most of the practically interesting cases: Lipschitz domains and domains with cone property are admissible, as we explain below. Assumption ŽA2. is needed only for the Robin condition; we do not use it for the Neumann condition, while for the Dirichlet condition no assumptions on D are needed, except boundedness of D. Assumptions ŽA. hold, for instance, if D is a Lipschitz domain D g C 0, 1 , that is, a domain whose boundary is locally the graph of a Lipschitz function, and the Lipschitz constant does not depend on the local patch of the boundary. Also the following cone property for D suffices for ŽA. to hold: For any p g D there exists a cone K Žwith the vertex at p . contained in D together with its closure, K [ x : < x9 < 2 F bx n2 , 0 - x n - a; b ) 0, x9 s Ž x 1 , . . . , x ny1 . 4 . The class of domains having the cone property is larger than the class of Lipschitz domains as defined above. Assumption ŽA. holds also if D g EV21 , the class of domains for which there exists a bounded extension operator E : H 1Ž D. ª H 1ŽR n. ,
Eu Ž p . s u Ž p . if p g D.
This class also contains the Lipschitz domains. If we deal with boundary condition Ž2N. then we only consider assumption ŽA1. on S. For the Dirichlet problem we can drop the whole assumption ŽA. and assume just that D is a compact domain. In w2, p. 243x a necessary and sufficient condition for ŽA1. to hold is given. In w1x the boundary-value problems for second order elliptic equations in bounded Lipschitz domains are studied. We prove in the present work, that for the class of compact domains satisfying condition ŽA. appropriate to the boundary condition, the solution to Ž1. ] Ž4., as defined above, exists and is unique. In the proof we will
332
RAMM AND RUIZ
use two well known results: LEMMA 1 Ž Rellich’s Type Lemma, see w3, p. 25x.. Let u be a solution of Ž D q K 2 . u s 0 in < x < ) R, such that H< x
2. UNIQUENESS Let us start with the definition of the solution for general domains. DEFINITION 1. We say that u solves the scattering problem if, for every r ) a, Ža. u g H , Žb. for all test functions w g H 1 Ž D9., vanishing near infinity, the following integral identity holds,
HD9 ž K Žc.
2
uw y quw y a i j u j wi dx q
/
HS s uwds s 0,
Ž 10 .
u satisfies Ž3. and Ž4..
If Neumann Ž2N. or Robin condition Ž2R. are imposed then H s H 1 Ž Dr .. For the Neumann boundary condition the boundary integral in Ž10. is dropped. If the Dirichlet condition Ž2D. is imposed then H s H˚ 1 Ž Dr . Ž H˚ 1 Ž Dr . is the space of functions in H 1 Ž Dr . vanishing on S ., the boundary integral in Ž10. is dropped, and the test function w has to be taken in H˚ 1 Ž D9..
333
UNIQUENESS THEOREMS
Condition Ž4. makes sense: from Ž7., Ž8., and the regularity results for the weak solutions of Ž1., it follows that ¨ is smooth in DaX . Clearly if D is a Lipschitz domain, then Ž10. implies Eq. Ž1. and the boundary condition Ž2.. If s s 0 then the last integral in Ž10. is absent. The easier case of the Dirichlet condition is left to the reader as an exercise. In this case, the last integral in Ž10. vanishes and w runs through the subspace of H 1 Ž DrX . of functions vanishing on S in the sense of the embedding theorem. Let us state the main result of this section. THEOREM 1. If D is compact and assumptions Ž6. ] Ž9. hold, then the solution to the scattering problem is unique. Proof. We first pass from the homogeneous equation Ž1. to the equivalent non-homogeneous one in order to eliminate u 0 from the asymptotic condition at infinity Ž3. and Ž4.. This passage is standard. Fix r 0 ) a, take h g C`ŽR., such that h9Ž r . G 0, hŽ r . s 1 if r ) r 0 q 1, hŽ r . s 0 if r - r 0 , and define W Ž x . s ¨ Ž x . q Ž1 y hŽ< x <.. u 0 Ž x .. Inserting u s W q hu 0 in Ž10., using Ž8., Ž9., and Green’s formula, we obtain
HD9 ž K
2
Ww y qWw y a i jWj wi q fw dx q
/
HS s Ww s 0,
Ž 11 .
where f [ Ž D q K 2 .Ž hu 0 . is a C` function supported in r 0 - < x < - r 0 q 1. Since W Ž x . s ¨ Ž x . if < x < ) r 0 q 1, W satisfied the radiation condition lim
H
rª` < x
< Wr y iKW < 2 ds s 0.
Ž 12 .
To prove the uniqueness of the solution it is enough to check that from Ž11. with f s 0, and Ž12., it follows that W s 0. To do this, we use the standard strategy w3x. From Ž12. we obtain lim rª0
½H Ž Sr
K 2 < W < 2 q < Wr < 2 . ds q iK
HS Ž WW y WW . ds r
r
r
5
s 0.
Ž 13 .
If we prove
HS Ž WW y WW . ds s 0, r
r
Ž 14 .
r
for r ) a, then Ž13. and Lemma 1 imply that W s 0 in R n _ Ba , and W s 0 in D9 from Lemma 2. To derive Ž14., take in Ž11., with f s 0, the test function w s WhŽŽ< x < y r 0 .re . [ Whe , r 0 ) a, where h g C`ŽR., h G 0, hŽ r . s 0 if r - 1r2 and hŽ r . s 1 if r ) 3r2.
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Subtract from expression Ž11. its complex conjugate. The real-valuedness of q, s , and K, and the assumptions on a i j , imply
HD9 ž W W y W W / Ž h . j
j
e
j dx
s 0.
We have Ž he . j s h9ŽŽ< x < y r 0 .re .Ž x jrŽ e < x <.., Ž1re . h9ŽŽ< x < y r 0 .re . ª d Ž< x < y r 0 ., and W g C`Ž< x < ) a.. Therefore we can take e ª 0 and get Ž14.. Note that in this argument assumption ŽA. is not used. Theorem 1 is proved. The following lemma will be used in Section 3. LEMMA 3. If W is a solution of the equation LW s 0 in R n, which satisfies radiation condition Ž12., then W s 0. Proof. As in the proof of Theorem 1, if suffices to prove Ž14.. We have W LW y W LW s 0. Integrate over Br and use Green’s formula to get 0s
HB Ž Lw W , W x y Lw W , W x . dx q HS Ž WW y WW . ds. r
r
r
r
Assumptions Ž6. and Ž8. imply that the first integral vanishes, we get Ž14., and Lemma 1 implies W s 0.
3. EXISTENCE THEOREM 2. If assumptions Ž A. and Ž6. ] Ž9. hold, then there exists a solution to the scattering problem and this solution is unique. As in Theorem 1, it is sufficient to prove the existence of a function W g H such that Ž11. and Ž12. hold with f supported in the annulus r 0 - < x < - r 0 q 1, r 0 ) a. We give the argument for the Neumann condition Ž s s 0., the case of the Robin condition is treated similarly. The idea of the proof is to reduce the scattering problem to a Fredholm-type equation without using integral equations, and to derive the existence of its solution from the uniqueness of the solution, which is a consequence of Theorem 1. We first prove an auxiliary result stated in Proposition 1, then describe the above reduction, and then complete the proof of Theorem 2. Let us pass to the auxiliary result. Consider the problem LV s h in R n ,
V satisfies the radiation condition Ž 4 . .
Ž 15 .
UNIQUENESS THEOREMS
335
Denote H 0 [ L20 Ž DR ., R ) r 0 q 1, the set of L2 Ž DR . functions vanishing near SR , and assume that supp h ; DR . PROPOSITION 1. Gi¨ en an arbitrary h g H 0 there exists a unique solution V of Ž15. such that V g Hl2o c and 5 V 5 b - `, b ) 1, where 5 V 5 2b [
HR
Ž 1 q < x <.
b
dx.
Proof of Proposition 1. From Ž6. ] Ž8. it follows that L is a symmetric, semibounded from below, operator in L2 with domain H02 , the set of H 2 ŽR n . functions vanishing near infinity. We can take its Friedrichs’ extension, also called L, defined on the dense subset of the domain of the quadratic form associated to L q C ) 0, which is H 1. From the self adjointness of L it follows that for e ) 0 the function Ve [ Ž L q i e .y1 h is well defined. The following lemma yields the conclusions of Proposition 1. LEMMA 4. We ha¨ e Ve ª V
as e ª 0 in Hl2o c and in 5 5 b ,
Ž 16 .
and V sol¨ es Ž 15 . .
Ž 17 .
We prove Lemma 4 in two steps. Step 1. Under the assumption sup 5 Ve 5 b - `,
Ž 18 .
0- e - e 0
the assertions Ž16. and Ž17. hold. Step 2. Under the hypothesis of Proposition 1, inequality Ž18. holds. If Ž18. holds, then for any ball BR , 5 Ve 5 L2 Ž B R . - C Ž R ., and therefore Ve converges weakly in L2 Ž BR . to some function V. We have LVe s yi e Ve q h, 5 LVe 5 L2 Ž B R . - C Ž R, h., and from the interior elliptic estimates it follows that 5 Ve 5 H 2 Ž BXR . - C Ž R, R9, h.; R9 - R. Therefore, for any R - a, 5 Ve 5 L2 Ž B R . - C Ž R ., Ve ª V in H s Ž BR ., s - 2. Using again Ž15., we conclude that LVe converges in L2 Ž BR ., and from the interior elliptic estimates we know that 5 Ve y Ve 9 5 H 2 Ž BXR . F C Ž 5 LVe y LVe 9 5 L2 Ž B R . q 5 Ve y Ve 9 5 L2 Ž B R . . .
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It follows that Ve ª V strongly in Hl2o c . We can pass to the limit in the equation and get LV s h. We now prove convergence in the norm 5 5 b and verify the radiation condition for V. In BRX , R ) a, Ve satisfies the equation Ž D q K 2 q i e .Ve s 0. Therefore, Green’s formula yields Ve Ž x . s
HS
x g BRX ,
Ve Ž s . Ž ge Ž x y s . . r y Ž Ve . r Ž s . ge Ž x y s . ds, R
Ž 19 . where ge is the fundamental solution ge Ž x . [
Ž ie q K 2 . < x<
Ž ny2 .r2
vn
Ž ny2.r2
2 HŽŽ1. ny2.r2 Ž i e q K .
ž
1r2
< x< ,
/
and we take I Ž i e q K 2 .1r2 ) 0. From Ž19. it follows that < Ve Ž x . < F
c < x<
Ž ny1.r2
,
if < x < ) R 0 , 0 - e - e 0 .
Ž 20 .
Since Ve ª V strongly in Hl2o c , we can pass to the limit in Ž19. and get VŽ x. s
HS
V Ž s . Ž g 0 Ž x y s . . r y Ž V . r Ž s . g 0 Ž x y s . ds,
x g BRX .
R
From this representation of V it follows that V satisfies the radiation condition. Finally, from estimate Ž20. and the pointwise convergence Ve ª V it follows that 5 Ve y V 5 b ª 0 and Step 1 is completed. Let us pass to Step 2. Arguing by contradiction, assume the existence of sequences e k ª 0, Vk [ Ve k, such that 5 Vk 5 b ª `. Since Ve s Ž L q i e .y1 h and 5 Ve 5 b F Ž1re .5 h 5 L2 , we can define Wk [ Vkr5 Vk 5 b . Then
Ž L q i e . Wk s
h 5 Vk 5 b
[ hk
and
5 h k 5 L2 Ž B R . ª 0.
Ž 21 .
Since 5 Wk 5 b s 1, under condition Ž21. we can repeat the arguments in Step 1 and prove that Wk ª W in Hl2o c , W satisfies the radiation condition, 5 Wk y W 5 b ª 0, and LW s 0. From the uniqueness Lemma 3 it follows that W s 0. This contradicts the relations 5 Wk y W 5 b ª 0 and 5 Wk 5 b s 1. Proposition 1 is proved.
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Let us now describe the reduction of the scattering problem in the form Ž11. ] Ž12. to a Fredholm-type equation Žw3, p. 36x.. Set W [ V y h Z,
Ž 22 .
where V solves Ž15. and h is a C` function which equals 1 near S and 0 outside DR . In particular, h s 0 on SR . The function W solves problem Ž11. ] Ž12. if Z solves the problem yf s h y L Ž h Z . in DR ,
GŽ Z. s GŽ V . ,
Z s 0 on SR , Ž 23 .
where we have used the strong formulation of the problem for convenience of the reader and with the understanding that Ž23. is understood in the weak sense, similarly to Ž11.. Note that near infinity W, defined in Ž22., equals V and, by the definition of V, satisfies the radiation condition Ž12.. There are many Z which solve Ž23. Žsince h in Ž23. is arbitrary .. Let us fix a unique Z as the solution to the problem LZ s iZ in DR ,
G Ž Z . s G Ž V . on S,
Z s 0 on SR ,
Ž 24 .
where again the weak formulation via the integral identity is understood. Clearly the solution to Ž24. is a linear operator on h. Define Bh [ LŽh Z .. Then Eq. Ž23. can be written as h y Bh s yf .
Ž 25 .
The operator B in Ž25. is compact in H 0 . Indeed, LŽh Z . s h LZ q QZ, where QZ [ LŽh Z . y h LZ contains not higher than the first derivatives of Z and LZ s iZ by Ž24.. The map h ª V ª Z ª QZ is the map H 0 ª H 1, as follows from the known estimates for the solutions of second order elliptic equations in bounded domains Žrecall that QZ vanishes near non-smooth boundary S because h s 1 near S .. By assumption ŽA1., the embedding H 0 ª H 1 is compact, so B is compact in H 0 . If the Robin boundary condition is used, then we use assumption ŽA2. also. Suppose that h solves Ž25., V solves Ž15., and W is defined by Ž22.. Then W solves Ž11. ] Ž12., as we checked above. Therefore, the scattering problem in the form Ž11. ] Ž12. is reduced to Eq. Ž25. with compact operator B in H 0 . The proof of Theorem 2 will be completed as soon as we show that the homogeneous version of Eq. Ž25. has only the trivial solution. Let us show this and thus complete the proof of Theorem 2. Assume that h solves Ž25. with f s 0. Then W, defined by Ž22., solves Ž11. ] Ž12. with f s 0. By Theorem 1, we get W s 0 in D9. Thus, V s h Z in DR , and V s Z on S. This and the first boundary condition Ž24. imply that V and Z have the same Cauchy data on S. Therefore, the function Z,
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extended into D so that Z s V in D, solves the problem LZ s iZ in DR ,
LZ s 0 in D,
Z s 0 on S R ,
Ž 26 .
where the equation LZ s 0 in D follows from Ž15. and the fact that h s 0 in D. Since L is symmetric, it follows from Ž26. that Z s 0 in BR . Therefore V s 0 in DR , and h s LV s 0 in DR . Theorem 2 is proved. Remark 1. We can relax condition Ž8.: allow for q g L lpo c , p ) nr2, and allow q s oŽ< x
REFERENCES 1. C. Kenig, ‘‘Harmonic Analysis Techniques for Second Order Elliptic Boundary Problems,’’ Amer. Math. Soc., Providence, Rhode Island, 1994. 2. V. Mazja, ‘‘Sobolev Spaces,’’ Springer-Verlag, New York, 1985. 3. A. G. Ramm, ‘‘Scattering by Obstacles,’’ Reidel, Dordrecht, 1986. 4. A. G. Ramm, Uniqueness theorems for inverse obstacle scattering problems in Lipschitz domains, Appl. Anal., 59, Ž1995., 377]383. 5. C. Wilcox, Scattering theory for the D’ Alembert equation in exterior domains, in ‘‘Lecture Notes in Math.,’’ Vol. 442, Springer-Verlag, New York, 1975.