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

This article is cited in 12 scientific papers (total in 12 papers)

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

Full text: PDF file (342 kB)
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English version:
Functional Analysis and Its Applications, 2009, 43:3, 165–183

Bibliographic databases:

UDC: 517.98+517.95
Received: 19.01.2009

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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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. M. S. Agranovich, “Spectral problems in Lipschitz domains”, Journal of Mathematical Sciences, 190:1 (2013), 8–33  mathnet  crossref  mathscinet
    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. 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
    5. Agranovich M.S., “Remarks on strongly elliptic systems in Lipschitz domains”, Russ. J. Math. Phys., 19:4 (2012), 405–416  crossref  mathscinet  zmath  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  crossref  mathscinet  zmath  isi
    8. V. S. Rabinovich, “Acoustic Diffraction Problems on Periodic Graphs”, Funct. Anal. Appl., 48:4 (2014), 298–303  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. V. Rabinovich, “Diffraction by periodic graphs”, Complex Var. Elliptic Equ., 59:4 (2014), 578–598  crossref  mathscinet  zmath  isi  elib
    10. V. Rabinovich, “Boundary problems for domains with conical exits at infinity and limit operators”, Complex Var. Elliptic Equ., 60:3 (2015), 293–309  crossref  mathscinet  zmath  isi
    11. Rabinovich V., “Integral Equations of Diffraction Problems With Unbounded Smooth Obstacles”, Integr. Equ. Oper. Theory, 84:2 (2016), 235–266  crossref  mathscinet  zmath  isi
    12. Rabinovich V., “Lp -theory of boundary integral operators for domains with unbounded smooth boundary”, Georgian Math. J., 23:4 (2016), 595–614  crossref  mathscinet  zmath  isi  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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