EIGENVALUES OF THE NEGATIVE LAPLACIAN FOR ARBITRARY MULTIPLY CONNECTED DOMAINS

The purpose of this paper is to derive some interesting asymptotic formulae for spectra of arbitrary multiply connected bounded domains in two or three dimensions, linked with 'variation of positive distinct functions entering the boundary conditions, using the spectral function


INTRODUCTION.
The underlying inverse eigenvalue problem (1.1)-(1.2) has been discussed recently by Zayed [1] and has shown that some geometric quantities associated with a bounded domain can be found from a complete knowledge of the eigenvalues {#k()}__ for the negative Laplacian A, (b-,) 2 in i=1 Rn(n=2or3).
Let ft be a simply connected bounded domain in R" with a smooth boundary Oft in the case n 2 (or a smooth bounding surface S in the case n 3).Consider the impedance problem -AnU=AU in ft (1.1) + a u 0 on On (or S), (1.2) wheredenotes differentiation along the inward pointing normal to Oft or S, and cr is a positive function.
(1.3)  where Ill[ and V are respectively the area and the volume of f.
The purpose of this paper is to discuss the following more general inverse problem: Let fl be bitr multiply coected bounded domain in R"(n 2 or 3) wch is suounded internally by simply coected bound domns , with smooth undes O, in the case n 2 (or smooth bounding surfaces S, in the case n 3) where 1, 2, m-1, d eemly by a simply co,coted bound domn fl th a smooth bounda 0 in the case n 2 (or a smooth bounding surface S in the case n 3).Suppose that the eigenvues 0 < (a,...,a) (a,...,a) (a,...,a) as k (1.6) c o exactly for the impedance problem u Au in , (1.7) ( + a, u 0 on 0fi (or Si) (1.$) where ,denote differentiations along the inward pointing normals to Off, or Si resptively md positive netions (i 1, m).
In Theorem 2.1, we detene some geometric qutities asiated th the multiply roman fl from the complete owlge of the eigenvues (1.6) for the problem (1.7)-(1.8)using the asptotie expsion of the setr netion 1 k(a], am) + p]2 as P , (1.9) where P is a positive nstt, wle i e positive nions defin on Ofli or Si (i 1, m) d tisng the Lipsctz ndition.
In Threm 2.2, we show that the asptotic expsion of (1.9) as P plays impot role in establisg a method to study the asptotic behaor of the difference (,..., m) ( ,flm)] A , (I.lO) 0<(a a)SA where a(Q), a(Q), B,(Q) Q Off, (or Q S), (i 1 m) e, generly spg, sfinct nions d tisng the Lipm condition d the suation is ten over vues of k for wch k (a, a) A.
Note that threms d roes of ts par cont her results s to thorn obtn rntly by Zay d Yours [2]. 2.