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2 Communications in Nonlinear Science and Numerical Simulation 13 (28) An inverse problem with data on the part of the boundary A.G. Ramm Mathematics epartment, Kansas State University, Manhattan, KS , USA Received 18 May 26; accepted 3 June 26 Available online 3 August 26 Abstract Let u t = $ 2 u q(x)u :¼ Lu in [,1), where R 3 is a bounded domain with a smooth connected boundary S, and q(x) 2 L 2 (S) is a real-valued function with compact support in. Assume that u(x,) =, u =ons 1 S, u = a(s,t) on S 2 = SnS 1, where a(s,t) = for t > T, a(s,t) 6, a 2 C 1 ([, T];H 3/2 (S 2 )) is arbitrary. Given the extra data u N j S2 ¼ bðs; tþ, for each a 2 C 1 ([, T];H 3/2 (S 2 )), where N is the outer normal to S, one can find q(x) uniquely. A similar result is obtained for the heat equation u t ¼ Lu :¼ rðaruþ. These results are based on new versions of Property C. Ó 26 Elsevier B.V. All rights reserved. MSC: 35K2, 35R3 PACS: 2.3.Jr Keywords: Property C; Parabolic equations; Inverse problems 1. Introduction Let R 3 be a bounded domain with a smooth connected boundary S, S = S 1 [ S 2, S 1 is an open subset in S, and S 2 is the complement of S 1 in S. Consider the problem u t ¼r 2 u qðxþu; ðx; tþ 2 ½; 1Þ; ð1þ uðx; Þ ¼; uj S1 ¼ ; uj S2 ¼ aðs; tþ; ð2þ where q(x) 2 L 2 () is a real-valued function, a(s,t) 2 C 1 ([,1), H 3/2 (S 2 )), the function a is an arbitrary realvalued function in the above set such that a 6, a = for t > T, H is the Sobolev space. For each a in the above set let the extra data be given: u N j S2 ¼ bðs; tþ; ð3þ where N is the outer normal to the boundary S. address: ramm@math.ksu.edu /$ - see front matter Ó 26 Elsevier B.V. All rights reserved. doi:1.116/j.cnsns

3 o the data {a(s,t), b(s,t)} "a(s,t) 2 C 1 ([,1), H 3/2 (S 2 )), a 6, a =for t > T, determine q(x) uniquely? Our main result is a positive answer to this question. Theorem 1. The data faðs; tþ; bðs; tþg 8tP;8s2S2, given for all a with the above properties, determine a compactly supported in real-valued function q(x) 2 L 2 () uniquely. Actually we prove a slightly stronger result: the data for 6 t 6 T + e determine q uniquely, where e > is an arbitrary small number. Note that the set S 2 can be arbitrary small. Theorem 1 is a multidimensional generalization of the author s result from [1]. Let and L j v kv :¼ r 2 v q j ðxþv kv; j ¼ 1; 2; N j ¼fv : L j v kv ¼ in; vj S1 ¼ g: The main tool in the proof of Theorem 1 is the following new version of Property C. Originally this property was introduced by the author in 1986 for the products of solutions to homogeneous linear partial differential equations in the case when these solutions did not satisfy any boundary conditions [3]. Theorem 2. The set {v 1,v 2 } for all v 1 2 N 1 and all v 2 2 N 2 is complete (total) in L 2 ( 1 ), where 1 is a strictly inner subdomain of, i.e., if p 2 L 2 ( 1 ), p = in n 1,and R 1 pv 1 v 2 dx ¼ "v 1 2 N 1, "v 2 2 N 2, then p =. In Section 2 Theorems 1 and 2 are proved. In Section 3 the results are generalized to the boundary-value problems for the equation u t = $ Æ (a(x)$u). 2. Proofs A.G. Ramm / Communications in Nonlinear Science and Numerical Simulation 13 (28) Proof of Theorem 1. Let v ¼ R 1 e kt udt and Aðs; kþ ¼ R 1 e kt aðs; tþdt. Taking the Laplace transform of the relations (1) and (2), we obtain: Lv kv ¼ ; Lv ¼r 2 v qðxþv; vj S1 ¼ ; vj S2 ¼ Aðs; kþ; ð4þ and v N ¼ Bðs; kþ ¼ R 1 e kt bðs; tþdt. Assume that there are q 1 and q 2, compactly supported in, which generate the same data. Let L j v j = $ 2 v j q j v j, j =1,2,andw = v 1 v 2. Then L 1 w kw ¼ pv 2 ; p ¼ q 1 q 2 ; ð5þ and for any w 2 N(L 1 k) :¼ {w: (L 1 k)w = } such that wj S1 ¼, one gets pv 2 w dx ¼ ðw N w w N wþds ¼ ; ð6þ S 2 because w = w =ons 1 and w = w N =ons 2 by our assumptions. By Theorem 2 relation (6) implies p =. Theorem 1 is proved. h Proof of Theorem 2. It is proved in [2] (see also [4]), that the set {v 1 v 2 } for all v j 2 M j :¼fv j : ðl j kþv j ¼ ing is total in L 2 (). Therefore it is sufficient to prove that N j is dense in M j in L 2 ( 1 )-norm, where 1 is a strictly inner subdomain of out of which both q 1 and q 2 vanish. Let us take j = 1. The proof for j = 2 is the same. Assume the contrary. Then there is a w 2 M 1 such that ¼ wvdx 8v 2 N 1 : ð7þ 1

4 536 A.G. Ramm / Communications in Nonlinear Science and Numerical Simulation 13 (28) Let G(x,y) solve the problem ðl 1 kþg ¼ dðx yþ in R 3 ; lim Gðx; yþ ¼; Gj S1 ¼ : ð8þ jx yj!1 Since Gðx; yþ 2N 1 8y 2 1 ¼ R3 n 1, Eq. (7) implies ¼ wðxþgðx; yþdx :¼ hðyþ; 8y 2 1 : ð9þ 1 We have hðyþ 2H 2 loc ðr3 n S 1 Þ, and h solves the elliptic equation ð kþh ¼ in 1 ; because q 1 = q 2 =in 1. Therefore, one gets from (9) the following relations: h ¼ h N ¼ on o 1 ; ð1þ and, because of (8), one gets ðl 1 kþh ¼ w in 1 : ð11þ Multiply (11) by w, integrate over 1, use (1), and get jwj 2 dx ¼ h N w hw N ds ¼ : ð12þ 1 o 1 Thus w =in 1 and, therefore, w =in, because w solves a homogeneous linear elliptic equation for which the uniqueness of the solution to the Cauchy problem holds. Theorem 2 is proved. h 3. Generalizations Consider now the problem u t ¼rðaðxÞruÞ :¼ Lu; ðx; tþ 2 ½; 1Þ; ð13þ uðx; Þ ¼; uj S1 ¼ ; uj S2 ¼ hðs; tþ; ð14þ and the extra data are au N j S2 ¼ zðs; tþ 8s 2 S 2 ; 8t > : ð15þ We assume that < a 6 a(x) 6 a 1, where a and a 1 are constants, a 2 H 3 (), h 2 C 1 ([,1), H 3/2 (S 2 )) is arbitrary, and prove the following theorem. Theorem 3. Under the above assumptions, the data fhðs; tþ; zðs; tþg 8s2S2 ;8t>, " h determine a(x) uniquely. Proof. Taking the Laplace transform of relations (13) (15), we reduce the problem to ðl kþv ¼ in; vj S1 ¼ ; vj S2 ¼ Hðs; kþ; av N j S2 ¼ ðs; kþ; ð16þ where, e.g., Hðs; kþ ¼ R 1 e kt hðs; tþdt. Assuming that a j, j = 1,2, generate the same data fhðs; kþ; ðs; kþg 8s2S2 ;8k>, one derives for w ¼ v 1 v 2 the problem ðl 1 kþw ¼rðprv 2 Þ; p :¼ a 2 a 1 ; wj S ¼ ; a 1 v 1N j S2 ¼ a 2 v 2N j S2 : ð17þ Multiply (17) by an arbitrary element of N 1, where N j ¼fu : ðl j kþu ¼ in; uj S1 ¼ g; j ¼ 1; 2; integrate by parts and get

5 prv 2 rudx þ pv 2N uds ¼ ða 1 w N u a 1 u N wþds: ð18þ S S Using boundary conditions (17), one gets prurv 2 dx ¼ ; 8u 2 N 1 8v 2 2 N 2 : ð19þ To complete the proof of Theorem 3, we use the following new version of Property C. Lemma 1. The set fru rv 2 g 8u2N1 ;8v 2 2N 2 is complete in L 2 () for all sufficiently large k >. We prove this lemma below, but first let us explain the claim made in the Introduction: Claim. The results remain valid if the data are given not for all t P but for t 2 [,T + e], where e > is arbitrarily small. This claim follows from the analyticity with respect to time of the solution u(x,t) to problems (1) and (2) and (13) and (14) in a neighborhood of the ray (T,1) for an arbitrary small e >. This analyticity holds if a(s,t) andh(s,t) vanish in the region t > T. Proof of Lemma 1. It was proved in [2, pp. 78 8], that the set frw 1 rw 2 g 8wj 2M j is complete in L 2 (), where M j ¼fw : ðl j kþw j ¼ g; L j kw ¼rða j rwþ kw; where k = const >, < c 6 a j (x) 6 C, a j (x) 2 H 3 (), and H () is the usual Sobolev space. Lemma 1 will be proved if we prove that any w j 2 M j can be approximated by an element v j 2 N j with an arbitrary small error in the norm, generated by the bilinear form ða j rw j ru þ kw j uþdx; u 2 H 1 ðþ: The above norm is equivalent to the norm of H 1 () due to the assumption < a 6 a(x) 6 a 1. Assuming that such an approximation is not possible, we can find a w j 2 M j such that ¼ a j ðxþrw j rg j ðx; yþþkw j G j ðx; yþ dx 8y 2 ; ð2þ because G j ðx; yþ 2N j for any y 2, G j solves (22) (see below) with G = G j. Integrating by parts in (2) and using the relation G j 2 N j, one gets ¼ a j w j G jn ðs; yþds :¼ hðyþ; 8y 2 : ð21þ S enote a j w j ¼ u; G j ¼ G; L j ¼ L: Since ðl kþg ¼ dðx yþ in R 3 n S 1 ; one can derive the relation lim G N ðs; yþ ¼ y!p2s 1 A.G. Ramm / Communications in Nonlinear Science and Numerical Simulation 13 (28) lim G ¼ ; and Gj S1 ¼ ; ð22þ jx yj!1 dðs pþ ; ð23þ aðsþ where d(s p) is the delta-function on S 1. We prove this relation later, but assuming that (23) holds we conclude from (21) that a j w j :¼ u ¼ on S 1 : ð24þ

6 538 A.G. Ramm / Communications in Nonlinear Science and Numerical Simulation 13 (28) Consequently hðyþ ¼ uðsþg N ðs; yþds ¼ S 2 8y 2 : ð25þ It follows from (25) that ðl kþh ¼ inr 3 n S 2 ; h ¼ in : ð26þ This and the uniqueness of the solution of the Cauchy problem for elliptic Eq. (26), imply h ¼ inr 3 n S 2 : ð27þ From (27) and the jump relation for the double-layer potential (25), we conclude that u ¼ on S 2 : ð28þ From (24) and (28) it follows that u =ons. Therefore w j =ons because a j >. Thus, w j 2 M j and w j = on S. This implies w j =in for all sufficiently large k >. Thus, Lemma 1 is proved if (23) is established. Let us prove (23). enote by g the (unique) solution to the problem ðl kþg ¼ dðx yþ in R 3 ; lim jx yj!1 gðx; yþ ¼: ð29þ Using Green s formula we obtain from (22) and (29) the relation Gðx; yþ ¼gðx; yþ gðx; sþaðsþg N ðs; yþds; S 1 x; y 2 S 1 :¼ R3 n S 1 : ð3þ Taking y! p 2 S 1 and using the boundary condition (22), we derive: gðx; pþ ¼lim gðx; sþaðsþg N ðs; yþds 8x 2 S y!p 1 : ð31þ S 1 Since the set fgðx; pþg 8x2S is dense in L 2 (S 1 ), formula (31) implies (23). 1 Let us finally check the claim that the set fgðx; pþg 8x2S is dense in L 2 (S 1 ). If it is not dense, then there is an 1 f 2 L 2 (S 1 ) such that ¼ f ðpþgðx; pþdp :¼ W ðxþ; S 1 8x 2 S 1 : ð32þ The integral in (32) is a simple-layer potential W(x) the density f(p) of which must vanish because of the jump formula for the normal derivatives of W across S 1. Thus the claim is proved. The proof of Lemma 1 is complete. Therefore Theorem 3 is proved. h Remark. If the conclusion of Lemma 1 remains valid for the set fru rv 2 g 8u2N1 ;v 2 2N 2, where v 2 is a single element of N 2, possibly chosen in a special way, then the conclusion of Theorem 1 will be established for the data which is a single pair of data faðs; tþ; bðs; tþg 8tP;8s2S2, where a(s,t) 6 is some function. References [1] Ramm AG. An inverse problem for the heat equation. J Math Anal Appl 21;264(2): [2] Ramm AG. Multidimensional inverse scattering problems. New York: Longman/Wiley; [3] Ramm AG. On completeness of the set of products of harmonic functions. Proc Am Math Soc 1986;99: [4] Ramm AG. Inverse problems. New York: Springer; 25.

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(This is a sample cover image for this issue. The actual cover is not yet available at this time.) This article appeared in a journal published by Elsevier. The attached copy is furnished to the author