Abstract:
In a bounded Lipschitz domain in $\mathbb{R}^n$, we consider a second-order
strongly elliptic system with symmetric principal part written in divergent form. We study the Neumann boundary value problem in a generalized variational (or weak) setting using the Lebesgue spaces $H^\sigma_p(\Omega)$ for solutions, where $p$ can differ from $2$ and $\sigma$ can differ from $1$. Using the tools of interpolation theory, we generalize the known theorem on the regularity of solutions, in which $p=2$ and $|\sigma-1|<1/2$, and the corresponding theorem on the unique solvability of the problem (Savaré, 1998) to $p$ close to $2$. We compare this approach with the nonvariational approach accepted in numerous papers of the modern theory of boundary value problems in Lipschitz domains. We discuss the regularity of eigenfunctions of the Dirichlet, Neumann, and
Poincaré–Steklov spectral problems.
Keywords:
second-order strongly elliptic system, Dirichlet, Neumann, and Poincaré–Steklov boundary value problems, variational solution, interpolation, regularity of solutions, Lebesgue and Besov spaces, regularity of eigenfunctions.
Citation:
M. S. Agranovich, “Regularity of Variational Solutions to Linear Boundary Value Problems in Lipschitz Domains”, Funktsional. Anal. i Prilozhen., 40:4 (2006), 83–103; Funct. Anal. Appl., 40:4 (2006), 313–329
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\by M.~S.~Agranovich
\paper Regularity of Variational Solutions to Linear Boundary Value Problems in Lipschitz Domains
\jour Funktsional. Anal. i Prilozhen.
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\pages 83--103
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Linking options:
https://www.mathnet.ru/eng/faa849
https://doi.org/10.4213/faa849
https://www.mathnet.ru/eng/faa/v40/i4/p83
This publication is cited in the following 16 articles:
Rabinovich V., “The Method of Potential Operators For Anisotropic Helmholtz Operators on Domains With Smooth Unbounded Boundaries”, Recent Trends in Operator Theory and Partial Differential Equations: the Roland Duduchava Anniversary Volume, Operator Theory Advances and Applications, 258, eds. Mazya V., Natroshvili D., Shargorodsky E., Wendland W., Springer International Publishing Ag, 2017, 229–256
Rabinovich V., “Integral Equations of Diffraction Problems With Unbounded Smooth Obstacles”, Integr. Equ. Oper. Theory, 84:2 (2016), 235–266
Rabinovich V., “Lp -theory of boundary integral operators for domains with unbounded smooth boundary”, Georgian Math. J., 23:4 (2016), 595–614
Rabinovich V., “Transmission Problems For Conical and Quasi-Conical At Infinity Domains”, Appl. Anal., 94:10 (2015), 2077–2094
Luisa Consiglieri, “Explicit Estimates for Solutions of Mixed Elliptic Problems”, International Journal of Partial Differential Equations, 2014 (2014), 1
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
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
Kozlov V., Nazarov S., “On the Hadamard Formula for Second Order Systems in Non-Smooth Domains”, Commun. Partial Differ. Equ., 37:5 (2012), 901–933
M. S. Agranovich, “Spectral problems in Lipschitz domains”, Journal of Mathematical Sciences, 190:1 (2013), 8–33
M. S. Agranovich, “Strongly Elliptic Second-Order Systems with Boundary Conditions on a Nonclosed Lipschitz Surface”, Funct. Anal. Appl., 45:1 (2011), 1–12
Volkwein S., “Admittance identification from point-wise sound pressure measurements using reduced-order modelling”, J. Optim. Theory Appl., 147:1 (2010), 169–193
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
Kozlov V., “Behavior of solutions to the Dirichlet problem for elliptic systems in convex domains”, Comm. Partial Differential Equations, 34:1 (2009), 24–51
M. S. Agranovich, “Spectral Boundary Value Problems in Lipschitz Domains for Strongly Elliptic Systems in Banach Spaces $H_p^\sigma$ and $B_p^\sigma$”, Funct. Anal. Appl., 42:4 (2008), 249–267
Agranovich M.S., “Remarks on potential spaces and Besov spaces in a Lipschitz domain and on Whitney arrays on its boundary”, Russ. J. Math. Phys., 15:2 (2008), 146–155
M. S. Agranovich, “To the Theory of the Dirichlet and Neumann Problems for Strongly Elliptic Systems in Lipschitz Domains”, Funct. Anal. Appl., 41:4 (2007), 247–263
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