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 Funktsional. Anal. i Prilozhen., 2009, Volume 43, Issue 3, Pages 3–25 (Mi faa2964)

Potential Type Operators and Transmission Problems for Strongly Elliptic Second-Order Systems in Lipschitz Domains

M. S. Agranovich

Moscow State Institute of Electronics and Mathematics

Abstract: We consider a strongly elliptic second-order system in a bounded $n$-dimensional domain $\Omega^+$ with Lipschitz boundary $\Gamma$, $n\ge2$. The smoothness assumptions on the coefficients are minimized. For convenience, we assume that the domain is contained in the standard torus $\mathbb{T}^n$. In previous papers, we obtained results on the unique solvability of the Dirichlet and Neumann problems in the spaces $H^\sigma_p$ and $B^\sigma_p$ without use of surface potentials. In the present paper, using the approach proposed by Costabel and McLean, we define surface potentials and discuss their properties assuming that the Dirichlet and Neumann problems in $\Omega^+$ and the complementing domain $\Omega^-$ are uniquely solvable. In particular, we prove the invertibility of the integral single layer operator and the hypersingular operator in Besov spaces on $\Gamma$. We describe some of their spectral properties as well as those of the corresponding transmission problems.

Keywords: strongly elliptic system, Lipschitz domain, Dirichlet problem, Neumann problem, Bessel potential space, Besov space, surface potential, transmission problem

DOI: https://doi.org/10.4213/faa2964

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English version:
Functional Analysis and Its Applications, 2009, 43:3, 165–183

Bibliographic databases:

UDC: 517.98+517.95

Citation: M. S. Agranovich, “Potential Type Operators and Transmission Problems for Strongly Elliptic Second-Order Systems in Lipschitz Domains”, Funktsional. Anal. i Prilozhen., 43:3 (2009), 3–25; Funct. Anal. Appl., 43:3 (2009), 165–183

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa2964
• https://doi.org/10.4213/faa2964
• http://mi.mathnet.ru/eng/faa/v43/i3/p3

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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. V. G. Maz'ya, M. Mitrea, T. O. Shaposhnikova, “The Inhomogeneous Dirichlet Problem for the Stokes System in Lipschitz Domains with Unit Normal Close to VMO”, Funct. Anal. Appl., 43:3 (2009), 217–235
2. M. S. Agranovich, “Spectral problems in Lipschitz domains”, Journal of Mathematical Sciences, 190:1 (2013), 8–33
3. M. S. Agranovich, “Strongly Elliptic Second-Order Systems with Boundary Conditions on a Nonclosed Lipschitz Surface”, Funct. Anal. Appl., 45:1 (2011), 1–12
4. M. S. Agranovich, “Mixed Problems in a Lipschitz Domain for Strongly Elliptic Second-Order Systems”, Funct. Anal. Appl., 45:2 (2011), 81–98
5. Agranovich M.S., “Remarks on strongly elliptic systems in Lipschitz domains”, Russ. J. Math. Phys., 19:4 (2012), 405–416
6. 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
7. Rabinovich V., “On Boundary Integral Operators for Diffraction Problems on Graphs with Finitely Many Exits at Infinity”, Russ. J. Math. Phys., 20:4 (2013), 508–522
8. V. S. Rabinovich, “Acoustic Diffraction Problems on Periodic Graphs”, Funct. Anal. Appl., 48:4 (2014), 298–303
9. V. Rabinovich, “Diffraction by periodic graphs”, Complex Var. Elliptic Equ., 59:4 (2014), 578–598
10. V. Rabinovich, “Boundary problems for domains with conical exits at infinity and limit operators”, Complex Var. Elliptic Equ., 60:3 (2015), 293–309
11. Rabinovich V., “Integral Equations of Diffraction Problems With Unbounded Smooth Obstacles”, Integr. Equ. Oper. Theory, 84:2 (2016), 235–266
12. Rabinovich V., “Lp -theory of boundary integral operators for domains with unbounded smooth boundary”, Georgian Math. J., 23:4 (2016), 595–614
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