Atlas: Spectral properties of potential type integral operators on smooth and Lipschitz surfaces by M. S. Agranovich

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Functional Analysis Valencia 2000
July 3-7, 2000
Technical University of Valencia (UPV) and University of Valencia (UV)
Valencia, Spain

Organizers
R.M. Aron (Kent State U., USA), K.D. Bierstedt (U. Paderborn, Germany), J. Bonet (UPV), J. Cerdà (U. Barcelona, Spain), H. Jarchow (U. Zürich, Switzerland), M. Maestre (UV), J. Schmets (U. Liège, Belgium)

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Spectral properties of potential type integral operators on smooth and Lipschitz surfaces
by
M. S. Agranovich
Moscow State Institute of Electronics and Mathematics (MGIEM), Moscow, Russia

Let S be a closed (n-1)-dimensional surface in Rⁿ, n >= 3. This surface divides its complement into two domains, bounded and unbounded. We consider a symmetric strongly elliptic second order system of partial differential equations in one or both of these domains. The coefficients are constant and real and satisfy some additional conditions. The system contains a frequency parameter which is a fixed number in the upper half-plane, and some corresponding conditions at infinity are imposed on solutions. We consider four spectral problems for this system with spectral parameter contained in boundary or transmission conditions on S. These problems are reduced to integral equations on S with the same spectral parameter, and our aim is to investigate spectral properties of the corresponding classical integral operators on S. The surface S is assumed to be either infinitely smooth (then they are elliptic pseudodifferential operators on S of order -1) or Lipschitz. We show that these operators are either self-adjoint operators or weak perturbations of self-adjoint operators, investigate the asymptotic behavior and the localization of eigenvalues, check the smoothness and prove the completeness and basic properties of the root functions in Sobolev L₂-spaces. For the Helmholtz equation, these problems have been proposed by Soviet physicists B.Z. Katsenelenbaum, N.N. Sivov, and N.N. Voitovich 30 years ago.

Especially difficult and interesting is the case of a Lipschitz surface, in which the results are of course weaker than in the case of a smooth surface. However, in the case of a smooth surface S the spectral properties of our integral operators are also of interest, and somewhat unexpected difficulties are encountered in the proofs. Complete proofs have been published up to now only in the cases of the Helmholtz equation and of Lamé system of isotropic elasticity.

(T)

Date received: November 30, 1999