HEARING THE SHAPE OF MEMBRANES : FURTHER RESULTS

The spectral function θ(t)=∑m=1∞exp(−tλm), t>0 where {λm}m=1∞ are the eigenvalues of the Laplacian in Rn, n=2 or 3, is studied for a variety of domains. Particular attention is given to circular and spherical domains with the impedance boundary conditions ∂u∂r


INTRODUCTION.
The underlying problems are to deduce the precise shape of membranes from the complete knowledge of the eigenvalues 0 < h < k2 < k3 <''" < km < as m , for the Laplace operator A in Rn, n 2 or 3. n (PI): Let R {(r,0): 0 < r < a, 0 < 0 < 27} be a circular domain of radius a and boundary r.
Suppose that the eigenvalues (I.I) are given for the eigenvalue equation (A 2 + %) u 0 in R together with the impedance boundary conditions: (-=--+ y.)u 0 on rj, J J, (1.2) where yj, J J are positive constants and the boundary Y consists of parts Y.I' j J such that rj {(r, 0): r a, aj o aj+I, J J, ctl= 0, aj+1= 2}.
The object of this paper is to determine the geometry of the domains in (PI) and (P2) as well as the impedances , J ,J from the asymptotic expansion of the spectral function (t) }. exp(-tm )' for small positive t.Zayed [I] has recently investigated probems (PI) and (P2) in the special case when J 2, that is, when the boundary r consists of two parts r I, r 2 and when the surface S consists of two parts $I, S 2. Finally, we close this introduction with the remark that the author [2,3] has recently generalized the results of [I] to the case R n when c n 2 or 3 is a simply connected bounded domain with a smooth boundary.
12 (2.7) HEARING THE SHAPE OF MEMBRANES: FURTHER RESULTS

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The problem now is to determine the asymptotic expansion of K(t) for small positive t.
In what follows we shall use Laplace transform with respect to "t" and use "s 2'' as the Laplace transform parameter; thus An application of the Laplace transform to the heat equation (2.2) shows 2 that G(x,x' ;s satisfies the two-dimenslonal membrane equation 2 together with the impedance boundary conditions (1.2).
The asymptotic expansion of K(t) as t 0, may then be deduced directly from the asymptotic expansion of (s 2) for s % where )axe. (

2.[01
With reference to section 3 in Stewartson and Waechter [5], it can readily be shown after some reduction that the impedance boundary conditions (1.2) give It now follows that the functions f.(v;s), J J may be expressed in terms of the asymptotic expansions of the modified Bessel functions and their derivatives due to Olver [6].These expansions for s are uniformly valid in u for arg u < .
Now, the following cases can be considered: CASE 1. (0 < rj << I, j j) In this case, it can be shown for s / that where T }+s2a 2 I/2 =V A ,n (T) fj (u;s)  Aj ,3 -T3( ayj+a yj )-(---+ 3ayj -a yj) TT( 2aY.i) + -z (2.15) J=l With reference to section in Zayed [I] and the articles by Kac [4], Gottlleb HEARING THE SHAPE OF MEMBRANES: FURTHER RESULTS 595 [7], Pleijel [8], and Steeman and Zayed [9], the asymptotic expansions (2.17), (2.19), (2.20) and (2.21) may be interpreted as: (1) is a circular domain of radius a and we have the impedance boundary conditions (1.2) with small/large impedances yj, j J as indicated in the specifications of the four respective cases, or (li) for the first three terms, is a bounded 2 domain in R 2 of area a Let h < be the number of smooth convex holes in J 3a j) holes and a boundary length of together with Dirlchlet boundary conditions.
We close this section with the remark that when J 2 the results (2.17), (2.19), (2.20) and (2.21) are in agreement with the results of [I].

CONSTRUCTION OF (t) FOR PROBLEM (P2).
In analogy with the two dimensional membrane problem, it is clear that t) associated with problem (P2) is given by: (t) G(,;t)d, (3.1)   where G(,';t) is the Green's function for the heat equation (A 3 -) u 0, (3.2) subject to the impedance boundary conditions (1.3) and the initial condition of the form (2.3).As we have done in section 2, we can write G(x,x';t) for problem (P2) in a form similar to (2.4), where 2 G0(,';t) (4t)   (3.7) With reference to section 2 in Waechter [I0], it can readily be shown after some reduction that the impedance boundary conditions (1.3) give 2 J (s 2) a2 [ (m + ) [ (a.j+l-aj) f.j(m;s)}, m--0 where fj(m;s) have the same form (2.12) with m replaced by m + The series (3.8) if fact diverges since K(t) for small positive t; however, this difficulty may be easily removed by considering the asymptotic expansion for large positive s of  With reference to section in [1] and the articles by Gottlleb [7], Waechter [I0], P1eiJel [11], and Zayed [12] the asymptotic expansions (3.13) (3.16) may be interpreted as (1) is a spherical domain of radius a and we have the impedance boundary conditions (1.3) with small/large impedances , j J as indicated in the specifications of the four respective cases, or (ii) for the first three 4 3 terms, is a bounded domain in R 3 of volume a 2 In case I, it has a surface S of area 4a the parts Sj, J I, J of the J surface S have areas 2a 2 ) .' (aj+ aj) and mean curvatures (;-3yj), jffil J j=l together with Neumann boundary conditions.
In case 2, the parts Sj, J I, k of the surface S have areas ;s) (I + --2 {Im(sa)Km(sa) a[sl'(sa) + 7. I (sa)] I(sa) + j Im(sa)] in which I m and K m are modified Bessel functions.The series (2.11) is slowly convergent for large positive s and it is therefore, expedient to apply a Watson transformation [
Neumann boundary conditions, provided h is an integer.k In case 2, it has h (al+l-.I)71 holes, the parts r], j k of the boundary 1" have lengths a ('+13 ') together with Neumann boundary conditions j=l J while the other parts rj, j k+l J have lengths -(j+l-j) (a*Y I) together with Dirlch]et boundary conditions, j=k+l JIn case 4, it has no holes (h 0) and a boundary length of ('+I -') fff (x,x;t)dx.
-aj) and mean curvatures (-3Vj) J k together with Neumann boundary conditions, while the other parts Sj, case 4, it has a surface of area 2a (0+ I-Ja4)(a-2yj and mean curvature j=l --together with Dirichlet boundary conditions.a E. M. E. ZAYED Finally, we note that when J 2 the results (3.13)(3.16)are in agreement with the results of [I].4.DISCUSSIONS.