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Spectral problems in Sobolev‐type Banach spaces for strongly elliptic systems in Lipschitz domains
Author(s) -
Agranovich M. S.
Publication year - 2016
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201500411
Subject(s) - mathematics , sobolev space , lipschitz continuity , lipschitz domain , bounded function , pure mathematics , banach space , hilbert space , type (biology) , resolvent formalism , resolvent , mathematical analysis , interpolation space , finite rank operator , functional analysis , ecology , biochemistry , chemistry , gene , biology
This paper is devoted to classical spectral boundary value problems for strongly elliptic second‐order systems in bounded Lipschitz domains, in general non‐self‐adjoint, namely, to questions of regularity and completeness of root functions (generalized eigenfunctions), resolvent estimates, and summability of Fourier series with respect to the root functions by the Abel–Lidskii method in Sobolev‐type spaces. These questions are not difficult in the Hilbert spaces of the typeH 1 = H 2 1 , and important results in this case are well‐known, but our aim is to extend the results to Banach spaces H p s with ( s , p ) in a neighborhood of (1, 2). We also touch upon some spectral problems on Lipschitz boundaries. Tools from interpolation theory of operators are used, especially the Shneiberg theorem.