THE WAVE EQUATION APPROACH TO THE TWO-DIMENSIONAL INVERSE PROBLEM FOR A GENERAL BOUNDED DOMAIN WITH PIECEWISE
SMOOTH MIXED BOUNDARY CONDITIONS

Abstract

Abstract:

The spectral distribution bμ(t) =P1!=1 exp(−itE12! ), where {E!}1!=1 are the eigenvalues of the negative Laplacian − = −P2=1( @ @x )2 in the (x1, x2)-plane, is studiedfor a variety of domains, where −1 < t < 1 and i = p−1. The dependence of bμ(t)on the connectivity of a domain and the boundary conditions are analyzed. Particular attention is given to a general bounded domain in R2 with a smooth boundary @ , where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth parts ??j (j = 1, · · · , n) of @ are considered such that @ = [nj=1??j .Some geometrical properties of (e.g., the area of , the total lengths of the boundary,the curvature of its boundary, etc.) are determined, from the asymptotic expansions of bμ(t) for |t| ! 0.

Key words: Inverse problem, spectral distribution, wave equation, eigenvalues.

CLC Number:

35K99

E.M.E.Zayed, I.H.Abdel-Halim. THE WAVE EQUATION APPROACH TO THE TWO-DIMENSIONAL INVERSE PROBLEM FOR A GENERAL BOUNDED DOMAIN WITH PIECEWISE
SMOOTH MIXED BOUNDARY CONDITIONS[J].Acta mathematica scientia,Series B, 2001, 21(2): 171-181.

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