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

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
\Bibitem{Agr11} \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. \mathnet{http://mi.mathnet.ru/cmfd171} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2830675} \transl \jour Journal of Mathematical Sciences \yr 2013 \vol 190 \issue 1 \pages 8--33 \crossref{https://doi.org/10.1007/s10958-013-1244-6} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84874946434} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. M. S. Agranovich, “Mixed Problems in a Lipschitz Domain for Strongly Elliptic Second-Order Systems”, Funct. Anal. Appl., 45:2 (2011), 81–98
3. Dmitrii K. Potapov, “Ob ellipticheskikh uravneniyakh so spektralnym parametrom i razryvnoi nelineinostyu”, Zhurn. SFU. Ser. Matem. i fiz., 5:3 (2012), 417–421
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
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
6. N. D. Kopachevsky, “Abstract Green formulas for triples of Hilbert spaces and sesquilinear forms”, Journal of Mathematical Sciences, 225:2 (2017), 226–264
7. A. A. Shkalikov, “Perturbations of self-adjoint and normal operators with discrete spectrum”, Russian Math. Surveys, 71:5 (2016), 907–964
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
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
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
11. Kostin A.B., “Carleman Parabola and the Eigenvalues of Elliptic Operators”, Differ. Equ., 54:3 (2018), 318–329
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