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Funktsional'nyi Analiz i ego Prilozheniya, 2008, Volume 42, Issue 4, Pages 2–23
DOI: https://doi.org/10.4213/faa2933
(Mi faa2933)
 

This article is cited in 15 scientific papers (total in 15 papers)

Spectral Boundary Value Problems in Lipschitz Domains for Strongly Elliptic Systems in Banach Spaces $H_p^\sigma$ and $B_p^\sigma$

M. S. Agranovich

Moscow State Institute of Electronics and Mathematics
References:
Abstract: In a bounded Lipschitz domain, we consider a strongly elliptic second-order equation with spectral parameter without assuming that the principal part is Hermitian. For the Dirichlet and Neumann problems in a weak setting, we prove the optimal resolvent estimates in the spaces of Bessel potentials and the Besov spaces. We do not use surface potentials. In these spaces, we derive a representation of the resolvent as a ratio of entire analytic functions with sharp estimates of their growth and prove theorems on the completeness of the root functions and on the summability of Fourier series with respect to them by the Abel–Lidskii method. Preliminarily, such questions for abstract operators in Banach spaces are discussed. For the Steklov problem with spectral parameter in the boundary condition, we obtain similar results. We indicate applications of the resolvent estimates to parabolic problems in a Lipschitz cylinder. We also indicate generalizations to systems of equations.
Keywords: strong ellipticity, Lipschitz domain, potential space, Besov space, weak solution, optimal resolvent estimate, determinant of a compact operator, completeness of root functions, Abel–Lidskii summability, parabolic semigroup.
Received: 28.05.2008
English version:
Functional Analysis and Its Applications, 2008, Volume 42, Issue 4, Pages 249–267
DOI: https://doi.org/10.1007/s10688-008-0039-x
Bibliographic databases:
Document Type: Article
UDC: 517.98+517.95
Language: Russian
Citation: M. S. Agranovich, “Spectral Boundary Value Problems in Lipschitz Domains for Strongly Elliptic Systems in Banach Spaces $H_p^\sigma$ and $B_p^\sigma$”, Funktsional. Anal. i Prilozhen., 42:4 (2008), 2–23; Funct. Anal. Appl., 42:4 (2008), 249–267
Citation in format AMSBIB
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  • This publication is cited in the following 15 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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