AN INVERSE EIGENVALUE PROBLEM FOR AN ARBITRARY MULTIPLY CONNECTED BOUNDED REGION IN R

The basic problem is to determine the geometry of an arbitrary multiply connected bounded region inR together with the mixed boundary conditions, from the complete knowledge ofthe eigenvalues {'}'-1 for the Laplace operator, using the asymptotic expansion ofthe spectral function 0(t) exp(-t) ast --0.


INTRODUCTION.
The underlying problem is to deduce the precise shape of a membrane from the complete knowledge of the eigenvalues {j}'.for the Laplace operator A2-i-i( .)2 in the xx2-plane.
The object of this paper is to discuss the following more general inverse problem: t be an arbitrary multiply connected bounded region in R which is surrounded internally by simply connected bounded domains , with smooth boundaries 0, m and externally by a simply connected bounded domain = with a smooth bounda #=.Suppose that the eigenvalues (1.3) are given for the eigenvalue equation (A+X)u=0 in , (1.7) together with one of the following mixed boundary conditions: 0u =0 on 0 i, i=l,...,k and u=0 on 0i, i=k+l,...,m, ( u-0 on 0,, i=l k and 0=0 on 0i, i=k+l m, (1.9) 0n where on denote differentiations along the inward inting normals to the boundaries 0, m, respectively.
The basic problem is to deteine the geomet of om the asymptotic expansion of e spectral function (1.4) for small sitive t.

STAMENT OF OUR ULTS.
Suppose that the boundaries 0, 1,...,m are given locally by the equations x" y'(o), n 1, 2 in which o, 1,...,m are the arc-lengths of the counterclocise oriented boundaries 0 and y"(o) C=(0).tL and k(o) be the lengths and the cuwatures of 0, m respectively.en, the results of our main problem (1.-(1.9)can be summarized in the following cases: CASE 1. .b.c. on 0i, 1 ,k and D.b.c. on 0,, k + 1 ,m) In this case the asymptotic expansion of 0(t) as 0 has the me form (2.1) with the interchanges 1, ...,k , k + 1 ,m. (2.1) With reference to formulae (1.4), (1.5)and to articles [6], [11], [12] the asymptotic expansion (2.1) may be interpreted as follows: (i) f is an arbitrary multiply connected bounded region in R and we have the mixed boundary conditions (1.8) or (1.9) as indicated in the specifications of the two respective cases.
(ii) For the first four terms, f is an arbitrary multiply connected bounded region in R:' of area f2 I . In
On setting xl-xx we find that O(t) + K(t), ( where The problem now is to determine the asymptotic expansion of K(t) for small positive t.In what follows we shall use Laplace transforms with respect to t, and use s as the Laplace transform parameter; thus we define (x_,,x_:;s:'-I ( R ) e -' 2 ' G ( x _ , , x _ 2 ; t ) d t .
The asymptotic expansion of K(t) for small positive t, may then be deduced directly from the asymptotic expansion of (s) for large positive s, where -(sz, ff(x_,x_;sm)dx_.It is well known [6] that the membrane equation (3.9) has the fundamental solution ) (s%) (4. )   where rx, x2 is the distance between the points x (x,x) and x (x,x) of the region f while K0 is the modified Bessel function of the second kind and of zero order.The existence of this solution enables us to construct integral equations for (xj,x_:;s2) satisfying the mixed boundary conditions (1.8)   or (1.9).Therefore, Green's theorem gives:  On the basis of (4.3)the function Xl,X,_;s'-) will be estimated for large values of s.The case when x and x,.lie in the neighborhoods of Og2,, m is particularly interesting.For this case, we need to use the following coordinates.
6. SOME LOCAL EXPANSIONS.It now follows that the local expansions of the functions when the distance between x and y is small, are very similar o those obtained in section 6 of [11].Con- sequently, for k,k + 1, ...,m, the local behavior of he following kernels: when the distance between y and y' is small, follows directly from the knowledge of the local epansions of(a.. DEFINITION 1.Let l and 2 be points in the upper half-plane j-0, hen we define )12 V(l 2l) 4-(21 4-)2. (6.4) An e ,;s -function is defined for points and belong to sufficiently small domains ?(I,) except when I, 1, ...,m and K is called the degree of this function.For every positive integer A i has the local expansion (see [11]): where " denotes a sum of a finite number of terms in which f(l) is an infinitely differentiable function.
In this expansion, P, P2, l, tn are integers, where Pl 0, P2 0, a: 0, g min(P + P:, q), q + m and the minimum is taken over all terms which occur in the summation Y.'.The remainderR^(t,2;s) has continuous derivatives of order d s A satisfying ,_2;s -0(s-Aes/''2) as s o, (6.6 where A is a positive constant. Thus, using methods similar to those obtained in section 7 of [11], we can show that the functions (6.1) are e X-functions with degrees k 0, -1 respectively.Consequently, the functions (6.2) are e-functions with degrees k-0,-1, while the functions (6.3) are eX-functions with degrees g 0,1 respectively.DEFINITION 2. IfXl andx2 are points in large domains f + 0fi, k,k + 1, ...,rn, then we define and f2-min(rxly+ry) if yE Ofi, i-1 ,k, J 2 min rx, + r if y _ Of2 An Ek(X_l,_x2;s)ofunction is defined and infinitely differentiable with respect tox 1_ and x 2_ when these points belong m large domains + 0fi except whenx x 0i, ,m.Thus, the E-function has a similar local expansion of the e-function (see [6], [11]).

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

First
Round of ReviewsMay 1, 2009 case 1, it has H (m 1) holes, the boundaries O,, k are of lengths L, and of curvatures , ...,k together with Neumann boundary conditions, while the boundaries Of2,, k + m are of lengths , L, and of curvatures k,(oi), k + m together with Dirichlet boundary conditions, i-k+l provided H is an integer.We close this section with the following remarks:REMARK 2.1.On setting k 0 in formula (2.1) with the usual definition that is zero, we obtain 1i-I the results of Dirichlet boundary conditions on Of2i, 1, ...,m.REMARK 2.2.On setting k m in formula (2.1) with the usual definition that is zero, we i-m+l obtain the results of Neumann boundary conditions on OQ,, 1, ...,m.