Journal of Mathematical Sciences

, Volume 190, Issue 1, pp 8–33 | Cite as

Spectral problems in Lipschitz domains

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Abstract

The paper is devoted to spectral problems for strongly elliptic second-order systems in bounded Lipschitz domains. We consider the spectral Dirichlet and Neumann problems and three problems with spectral parameter in conditions at the boundary: the Poincaré–Steklov problem and two transmission problems. In the style of a survey, we discuss the main properties of these problems, both self-adjoint and non-self-adjoint. As a preliminary, we explain several facts of the general theory of the main boundary value problems in Lipschitz domains. The original definitions are variational. The use of the boundary potentials is based on results on the unique solvability of the Dirichlet and Neumann problems. In the main part of the paper, we use the simplest Hilbert L 2-spaces H s , but we describe some generalizations to Banach spaces H s p of Bessel potentials and Besov spaces B s p at the end of the paper.

Keywords

Dirichlet Problem Elliptic System Spectral Problem Besov Space Neumann Problem 
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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Moscow Institute of Electronics and MathematicsMoscowRussia

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