AN INVERSE EIGENVALUE PROBLEM FOR AN ARBITRARY MULTIPLY CONNECTED BOUNDED REGION : AN EXTENSION TO HIGHER DIMENSIONS

The basic problem in this paper is that of detemnining the geometry of an arbitrary multiply connected bounded region in R3 together with the mixed boundary conditions, from the complete knowledge of the eigenvalues {,}= for the negative Laplacian, using the asymptotic expansion of the spectral function O(t)= ezp(tA) as t-,O.

In terms of the mean curvature H and Gaussian curvature N (1.7) may be rewritten --, in the forms: {' f(n N)es, in the case of D.b.c., The object of this paper is to discuss the following more general inverse problem: Let fl be an arbitrary multiply connected bounded region in R 3 which is surrounded internally by simply Note that problem (1.)-(1.11) has been investigated recently by Zayed [11] in the special case when f is an arbitrary doubly connected region (i.e., m 2). , [2], [7], [11], the a.symptotic expansion (2.1) may be interpreted as follows: (i) fl is an arbitrary multiply connected bounded region in R 3 and we have the mixed boundary conditions (1.10) or (1.11) as indicated in the specifications of the two respective cases.
(if) For the first five terms, fl is an arbitrary multiply connected bounded region in R 3 of volume V. In

FORMULATION OF THE MATHEMATICAL PROBLEM.
In analogy with the two-dimensional problem (soe tg, 10]), it is easy to show that associated with problem (1.9)-(1. On setting we find that O(t) (4rt)3/2 + K(t), (3.6) where In what follows, we shall use Laplace transforms with respect to t, and use as the Laplace transform parameter; thus we define An application of the Laplace transform to the heat equation (3.2) shows that i ,,2;s 2) satisfies the membraue equation A s-s)( t,L 2;s2) -/i -2) in , (3.9) together with the mixed boundary conditions (1.10) or (1.11).
The ymptotic expsion of K(t) s t0, may then deduced directly from the ymptotic  On applying the iteration method (see. [7], [9], [11]) to the integral equation  in Section 5.2 of [11] with the interchanges (li) and 6 6i, (i k). Thus, we have the same formulae (5.2.1)-(5.2.5) of Section 5.1 in [11] with the interchanges n n,, n (%) N,, (i k + l, .m) 6. SOME LOCAL EXPANSIONS. It now follows that the local expansions of the functions Fxf ,i=1 m, (6.1) when the distance between and y is small are very similar to those obtained in Section 6 of [11]. Consequently, the local behavior of the kernels ',(y ", ),''_,( ",y ), (6.2) g,(g ",g ), '_,(g" g ), (6.3) when the distance between y and y'is small, follows directly from the local expansions of the functions (6. ). An Ex ,;)-function is defined and infinitely differentiable with respect to and when these points belong to large domains fl + $i except when 2 Si, m. Thus, the r-ftmctioa ha a similar local expansion of the -fuaction (see [7], [11]).
With the help of Section 8 in [11], it is easily seen that formula (4.3) is an E-1, ;)function and consequently which is valid for s--.oo, where Ai(i m) are positive constants. Formula (6.4) shows that 0 ,2,s) is exponentially small for With reference to Sections 7 and 9 in [11], if the -expansions of the functions (6.1)-(6.3) are introduced into (4.3) and if we use formulae similar to (7.4) and (7.10) of Section 7 in [11], we obtain the following local behavior of [ , ;2) as -.oo which is valid when and are (6.5) where, if , and belong to sufficiently small domains (1,)  On inserting (7.2) into (3.6) we arrive at our result (2.1).