A uniqueness theorem for a boundary inverse problem

Let DcR' be a bounded domain with a smooth boundary r, A u + q ( x ) u = O in D u = f , u N = h on Iand ~ ( x ) E L ~ ( D ) . From knowledge of the set {f, h} where f runs through C'(r) the coefficient 9(x) is uniquely recovered. Analytical formulae for y(x) are given. Applications are considered. Let D c R3 be a bounded domain with a smooth boundary r, q(x) E L"(D) , A u + q ( x ) u = O in D U = f , U N = h On (1) where A is the Laplacian and N is the outward normal to r. Assume that zero is not an eigenvalue of the Dirichlet operator A + q in D . If q and f are known then U is uniquely defined in D . Therefore h is uniquely defined. We are interested in the inverse problem A: given the set of pairs {f, h} forfrunning through C'(T), determine Problems of this type were studied recently (see [l, 21 and references therein). In [ l ] the uniqueness of the solution to the inverse problem B of finding a(x) E C"(D), a(x) > 0 in D, D is the closure of D, from the set {fi,, ah,)}, where V (a (x )Vw) = 0 in D , w=fo, wN=h() on r, is proved. Some additional results are obtained in [2]. Our purpose is to show that some uniqueness theorems for inverse problems can be easily obtained by the method given in [3,4]. As an example we give a short (but nevertheless complete) proof of the following. 4x1 . Theorem I . The set {f, h} where f runs through C'(T) determines q(x) E L"(D) uniquely. Remark I . I t is known that problems A and B are closely related: put W = ( T ' ~ ? ( X ) U then U solves (1) with q(x) = a-"*Aa"* , If a and aN are known on r then knowledge of the set { f o , ah,)} implies knowledge of the set {f, h}. In applications problem B is the problem of finding the conductivity of a body from measurements of the potential and current on its surface. Remark 2. Some uniqueness theorems in inverse problems of geophysics are given in [5]. The ideas in [ l ] are applied to some inverse scattering problems in [6]. 0266-5611/88/010001+ 05 $02.50 01988 IOP Publishing Ltd L1 L2 Letter to the Editor Proof of theorem 1. We will use the result which is a particular case of proposition 1 from [4]: there exists a solution to equation (1) of the form u(x, z) = exp(iz x ) (1 + R(x , 2)) (2) llR(x, z) l lI ,2(D) Cif I-’” as IzI+m, z z = O . (3) where ZEC, z.z :=zf+zi+z:=O, and Here (21 = (lzll’+ 1 . ~ ~ 1 ’ + 1 ~ ~ 1 ’ ) ~ ’ ~ and c = constant >O; c does not depend on z but depends on IlqllLr(D). A similar result was first proved in [l]. Let 1~@3, L . A = O . We have exp(iA.x)qudw= exp(i2-x)Audx JD JD a a N = 1 (exp(i2 x)h f exp(i2x ) (4) where F is known by assumption. Substitute (2) into (4) and let IzI+ CO while keeping the following conditions satisfied: A * A = O z . z = o a + z = p ( 5 ) where p E R3 is an arbitrary given vector. Note that ( 5 ) are ten equations for twelve parameters: 1 and z are each determined by six real numbers. It is not difficult to check that conditions ( 5 ) can be satisfied (for any given P E R3 and Iz(+w). This is done in detail in [4]. If IzI+m, by (3), the left-hand side of (4) becomes J q ( x ) exp(ip x ) dw : =Lib). Since the right-hand side F of (4) is known we found 4(p) from the data. Namely, by the assumption the pairs {f, h} corresponding to the solutions (2) with any Z E @ ~ , z z=O, are known, so that Fin (4) is known. If q ( p ) is found, q(x) is obtained uniquely and effectively by taking the inverse Fourier transform. Theorem 1 is proved. Corolfaryl. If a a n d a N o n r a r e k n o w n , O < c s a ( x ) , x E D a n d o ( x ) E W’.=(D), then problem B has at most one solution. Furthermore, this solution can be effectively constructed as follows: (i) solve problem A with q(x) = u ” ~ A ~ ” * (see remark 1); (ii) find a(x) by solving the problem Aa”* q(x)a”* = 0 in D a’” and (o”’)~ are known on r. This is a Cauchy problem for the elliptic equation (6). By the uniqueness of the solution of the Cauchy problem for this equation one concludes that a”* is uniquely determined. For the compatible Cauchy data (7) all2 can actually be found as the solution to the Dirichlet problem Ad’* q(x)a”* = 0 in D a’’* is known on r. (8) Remark 3. Problem B has been discussed in [8] and it was proved that the set { fo, ah,,} determines a and all its derivatives on r provided that a E C”(D) and r E C ” . Letter to the Editor L3 Remark 4. One of the results in [6] is the following 'n-dimensional Borg-Levinson theorem': let q, (x) E C"(D) , Im q, = 0, j = 1,2, and suppose that A;) =A:), and q(l) mNon for all m = 1,2, .... Then q l ( x ) = q 2 ( x ) in D. Here ( A + q, (x) A$))+$) = 0 in D q $ ) = O on r a V m qmN:=on r. aN Some general results of this type are announced in [7]. One can use theorem 1 to prove such a result. llq!??llLqD) = 1 For example, if AG + q(x)G = d(x y) in D, G = 0 on r, then