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

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

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English version:
Functional Analysis and Its Applications, 2008, 42:4, 249–267

Bibliographic databases:

UDC: 517.98+517.95

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

Citation in format AMSBIB
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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
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
3. V. B. Shakhmurov, “Maximal regular abstract elliptic equations and applications”, Siberian Math. J., 51:5 (2010), 935–948
4. M. S. Agranovich, “Spectral problems in Lipschitz domains”, Journal of Mathematical Sciences, 190:1 (2013), 8–33
5. Agarwal R., O'Regan D., Shakhmurov V., “Degenerate anisotropic differential operators and applications”, Bound. Value Probl., 2011, 268032, 27 pp.
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
7. Shakhmurov V., “Estimates of approximation numbers and applications”, Acta Math. Sin. (Engl. Ser.), 28:9 (2012), 1883–1896
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
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
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
11. Wei W., Zhang Zh., “l-P Resolvent Estimates For Variable Coefficient Elliptic Systems on Lipschitz Domains”, Anal. Appl., 13:6 (2015), 591–609
12. Shakhmurov V.B., “Linear and Nonlinear Abstract Differential Equations of High Order”, Open Math., 13 (2015), 471–486
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
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