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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

 Funktsional. Anal. i Prilozhen., 2011, Volume 45, Issue 1, Pages 1–15 (Mi faa3031)

Strongly Elliptic Second-Order Systems with Boundary Conditions on a Nonclosed Lipschitz Surface

M. S. Agranovich

Moscow Institute of Electronics and Mathematics

Abstract: We consider boundary value problems and transmission problems for strongly elliptic second-order systems with boundary conditions on a compact nonclosed Lipschitz surface $S$ with Lipschitz boundary. The main goal is to find conditions for the unique solvability of these problems in the spaces $H^s$, the simplest $L_2$-spaces of the Sobolev type, with the use of potential type operators on $S$. We also discuss, first, the regularity of solutions in somewhat more general Bessel potential spaces and Besov spaces and, second, the spectral properties of problems with spectral parameter in the transmission conditions on $S$, including the asymptotics of the eigenvalues.

Keywords: strong ellipticity, Lipschitz domain, nonclosed boundary, potential type operators, Bessel potential spaces, Besov spaces, regularity of solutions, spectral transmission problems, spectral asymptotics

DOI: https://doi.org/10.4213/faa3031

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English version:
Functional Analysis and Its Applications, 2011, 45:1, 1–12

Bibliographic databases:

UDC: 517.98+517.95

Citation: M. S. Agranovich, “Strongly Elliptic Second-Order Systems with Boundary Conditions on a Nonclosed Lipschitz Surface”, Funktsional. Anal. i Prilozhen., 45:1 (2011), 1–15; Funct. Anal. Appl., 45:1 (2011), 1–12

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa3031
• https://doi.org/10.4213/faa3031
• http://mi.mathnet.ru/eng/faa/v45/i1/p1

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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, “Spectral problems in Lipschitz domains”, Journal of Mathematical Sciences, 190:1 (2013), 8–33
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. M. S. Agranovich, A. M. Selitskii, “Fractional Powers of Operators Corresponding to Coercive Problems in Lipschitz Domains”, Funct. Anal. Appl., 47:2 (2013), 83–95
4. Shlapunov A. Tarkhanov N., “On completeness of root functions of Sturm-Liouville problems with discontinuous boundary operators”, J. Differential Equations, 255:10 (2013), 3305–3337
5. P. Exner, K. Pankrashkin, “Strong coupling asymptotics for a singular Schrödinger operator with an interaction supported by an open arc”, Comm. Partial Differential Equations, 39:2 (2014), 193–212
6. N. Tarkhanov, A. A. Shlapunov, “Sturm–Liouville problems in weighted spaces in domains with nonsmooth edges. II”, Siberian Adv. Math., 26:4 (2016), 247–293
7. Shlapunov A., Peicheva A., “on the Completeness of Root Functions of Sturm-Liouville Problems For the Lame System in Weighted Spaces”, ZAMM-Z. Angew. Math. Mech., 95:11 (2015), 1202–1214
8. Sybil Yu.M., Grytsko B.E., “Boundary Value Problem For the Two-Dimensional Laplace Equation With Transmission Condition on Thin Inclusion”, J. Numer. Appl. Math., 2:122 (2016), 120–129
9. Peicheva A.S., “Embedding Theorems For Functional Spaces Associated With a Class of Hermitian Forms”, J. Sib. Fed. Univ.-Math. Phys., 10:1 (2017), 83–95
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