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CMFD, 2011, Volume 39, Pages 11–35 (Mi cmfd171)  

This article is cited in 11 scientific papers (total in 11 papers)

Spectral problems in Lipschitz domains

M. S. Agranovich

Moscow Institute of Electronics and Mathematics

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.

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English version:
Journal of Mathematical Sciences, 2013, 190:1, 8–33

Bibliographic databases:

UDC: 517.984.5

Citation: M. S. Agranovich, “Spectral problems in Lipschitz domains”, Partial differential equations, CMFD, 39, PFUR, M., 2011, 11–35; Journal of Mathematical Sciences, 190:1 (2013), 8–33

Citation in format AMSBIB
\by M.~S.~Agranovich
\paper Spectral problems in Lipschitz domains
\inbook Partial differential equations
\serial CMFD
\yr 2011
\vol 39
\pages 11--35
\publ PFUR
\publaddr M.
\jour Journal of Mathematical Sciences
\yr 2013
\vol 190
\issue 1
\pages 8--33

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    This publication is cited in the following articles:
    1. M. S. Agranovich, “Strongly Elliptic Second-Order Systems with Boundary Conditions on a Nonclosed Lipschitz Surface”, Funct. Anal. Appl., 45:1 (2011), 1–12  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. M. S. Agranovich, “Mixed Problems in a Lipschitz Domain for Strongly Elliptic Second-Order Systems”, Funct. Anal. Appl., 45:2 (2011), 81–98  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. Dmitrii K. Potapov, “Ob ellipticheskikh uravneniyakh so spektralnym parametrom i razryvnoi nelineinostyu”, Zhurn. SFU. Ser. Matem. i fiz., 5:3 (2012), 417–421  mathnet
    4. Alexander N. Polkovnikov, Aleksander A. Shlapunov, “On the spectral properties of a non-coercive mixed problem associated with $\overline\partial$-operator”, Zhurn. SFU. Ser. Matem. i fiz., 6:2 (2013), 247–261  mathnet
    5. Shlapunov A., Tarkhanov N., “On Completeness of Root Functions of Sturm-Liouville Problems with Discontinuous Boundary Operators”, J. Differ. Equ., 255:10 (2013), 3305–3337  crossref  mathscinet  zmath  isi  elib  scopus
    6. N. D. Kopachevsky, “Abstract Green formulas for triples of Hilbert spaces and sesquilinear forms”, Journal of Mathematical Sciences, 225:2 (2017), 226–264  mathnet  crossref
    7. A. A. Shkalikov, “Perturbations of self-adjoint and normal operators with discrete spectrum”, Russian Math. Surveys, 71:5 (2016), 907–964  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. V. N. Pavlenko, D. K. Potapov, “Existence of two nontrivial solutions for sufficiently large values of the spectral parameter in eigenvalue problems for equations with discontinuous right-hand sides”, Sb. Math., 208:1 (2017), 157–172  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. Anastasiya S. Peicheva, “Embedding theorems for functional spaces associated with a class of Hermitian forms”, Zhurn. SFU. Ser. Matem. i fiz., 10:1 (2017), 83–95  mathnet  crossref
    10. O. A. Andronova, V. I. Voytitskiy, “On spectral properties of one boundary value problem with a surface energy dissipation”, Ufa Math. J., 9:2 (2017), 3–16  mathnet  crossref  isi  elib
    11. Kostin A.B., “Carleman Parabola and the Eigenvalues of Elliptic Operators”, Differ. Equ., 54:3 (2018), 318–329  crossref  mathscinet  zmath  isi  scopus
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