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Funktsional. Anal. i Prilozhen., 2008, Volume 42, Issue 4, Pages 2–23 (Mi faa2933)  

This article is cited in 13 scientific papers (total in 13 papers)

Spectral Boundary Value Problems in Lipschitz Domains for Strongly Elliptic Systems in Banach Spaces $H_p^\sigma$ and $B_p^\sigma$

M. S. Agranovich

Moscow State Institute of Electronics and Mathematics

Abstract: In a bounded Lipschitz domain, we consider a strongly elliptic second-order equation with spectral parameter without assuming that the principal part is Hermitian. For the Dirichlet and Neumann problems in a weak setting, we prove the optimal resolvent estimates in the spaces of Bessel potentials and the Besov spaces. We do not use surface potentials. In these spaces, we derive a representation of the resolvent as a ratio of entire analytic functions with sharp estimates of their growth and prove theorems on the completeness of the root functions and on the summability of Fourier series with respect to them by the Abel–Lidskii method. Preliminarily, such questions for abstract operators in Banach spaces are discussed. For the Steklov problem with spectral parameter in the boundary condition, we obtain similar results. We indicate applications of the resolvent estimates to parabolic problems in a Lipschitz cylinder. We also indicate generalizations to systems of equations.

Keywords: strong ellipticity, Lipschitz domain, potential space, Besov space, weak solution, optimal resolvent estimate, determinant of a compact operator, completeness of root functions, Abel–Lidskii summability, parabolic semigroup

DOI: https://doi.org/10.4213/faa2933

Full text: PDF file (331 kB)
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English version:
Functional Analysis and Its Applications, 2008, 42:4, 249–267

Bibliographic databases:

UDC: 517.98+517.95
Received: 28.05.2008

Citation: M. S. Agranovich, “Spectral Boundary Value Problems in Lipschitz Domains for Strongly Elliptic Systems in Banach Spaces $H_p^\sigma$ and $B_p^\sigma$”, Funktsional. Anal. i Prilozhen., 42:4 (2008), 2–23; Funct. Anal. Appl., 42:4 (2008), 249–267

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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, “Potential Type Operators and Transmission Problems for Strongly Elliptic Second-Order Systems in Lipschitz Domains”, Funct. Anal. Appl., 43:3 (2009), 165–183  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. V. B. Shakhmurov, “Maximal regular abstract elliptic equations and applications”, Siberian Math. J., 51:5 (2010), 935–948  mathnet  crossref  mathscinet  isi  elib
    4. M. S. Agranovich, “Spectral problems in Lipschitz domains”, Journal of Mathematical Sciences, 190:1 (2013), 8–33  mathnet  crossref  mathscinet
    5. Agarwal R., O'Regan D., Shakhmurov V., “Degenerate anisotropic differential operators and applications”, Bound. Value Probl., 2011, 268032, 27 pp.  mathscinet  zmath  isi  elib
    6. 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
    7. Shakhmurov V., “Estimates of approximation numbers and applications”, Acta Math. Sin. (Engl. Ser.), 28:9 (2012), 1883–1896  crossref  mathscinet  zmath  isi
    8. 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  adsnasa  isi  elib
    9. Wei W., Zhang Zh., “L-P Resolvent Estimates For Constant Coefficient Elliptic Systems on Lipschitz Domains”, J. Funct. Anal., 267:9 (2014), 3262–3293  crossref  mathscinet  zmath  isi
    10. 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  mathnet  crossref  crossref  mathscinet  elib
    11. Wei W., Zhang Zh., “l-P Resolvent Estimates For Variable Coefficient Elliptic Systems on Lipschitz Domains”, Anal. Appl., 13:6 (2015), 591–609  crossref  mathscinet  zmath  isi  elib
    12. Shakhmurov V.B., “Linear and Nonlinear Abstract Differential Equations of High Order”, Open Math., 13 (2015), 471–486  crossref  mathscinet  zmath  isi  elib
    13. Agranovich M.S., “Spectral problems in Sobolev-type Banach spaces for strongly elliptic systems in Lipschitz domains”, Math. Nachr., 289:16 (2016), 1968–1985  crossref  mathscinet  zmath  isi  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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