Abstract:
The classical solution of the Dirichlet problem with a continuous boundary function for a linear elliptic equation with Hölder continuous coefficients and right-hand side satisfies the interior Schauder estimates describing the possible increase of the solution smoothness characteristics as the boundary is approached, namely, of the solution derivatives and their difference ratios in the corresponding Hölder norm. We prove similar assertions for the generalized solution with some other smoothness characteristics. In contrast to the interior Schauder estimates for classical solutions, our established estimates for the differential characteristics imply the continuity of the generalized solution in a sense natural for the problem (in the sense of (n−1)-dimensional continuity) up to the boundary of the domain in question. We state the global properties in terms of the boundedness of the integrals of the square of the difference between the solution values at different points with respect to especially normalized measures in a certain class.
Keywords:
elliptic equation, smoothness of solution, function space.
Citation:
A. K. Gushchin, “A strengthening of the interior Hölder continuity property for solutions of the Dirichlet problem for a second-order elliptic equation”, TMF, 157:3 (2008), 345–363; Theoret. and Math. Phys., 157:3 (2008), 1655–1670
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\paper A~ strengthening of the~interior H\"older continuity property for solutions of the~Dirichlet problem for a~second-order elliptic equation
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\yr 2008
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\issue 3
\pages 345--363
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\jour Theoret. and Math. Phys.
\yr 2008
\vol 157
\issue 3
\pages 1655--1670
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Linking options:
https://www.mathnet.ru/eng/tmf6284
https://doi.org/10.4213/tmf6284
https://www.mathnet.ru/eng/tmf/v157/i3/p345
This publication is cited in the following 12 articles:
A. G. Losev, E. A. Mazepa, “Asymptotic behavior of solutions of the Dirichlet problem for the Poisson equation on model Riemannian manifolds”, Sib. elektron. matem. izv., 19:1 (2022), 66–80
K. A. Bliznyuk, E. A. Mazepa, “Kraevye i vneshnie kraevye zadachi dlya uravneniya Puassona na nekompaktnykh rimanovykh mnogoobraziyakh”, Materialy Voronezhskoi mezhdunarodnoi zimnei matematicheskoi shkoly «Sovremennye metody teorii funktsii i smezhnye problemy», Voronezh, 28 yanvarya – 2 fevralya 2021 g. Chast 2, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 207, VINITI RAN, M., 2022, 3–9
A. K. Gushchin, “On some properties of elliptic partial differential equation solutions”, Int. J. Mod. Phys. A, 37:20 (2022), 2243002–9
A. K. Gushchin, “The boundary values of solutions of an elliptic equation”, Sb. Math., 210:12 (2019), 1724–1752
A. K. Gushchin, “On the Existence of L2 Boundary Values of Solutions to an Elliptic Equation”, Proc. Steklov Inst. Math., 306 (2019), 47–65
A. K. Gushchin, “A criterion for the existence of Lp boundary values of solutions to an elliptic equation”, Proc. Steklov Inst. Math., 301 (2018), 44–64
A. K. Guschin, “O zadache Dirikhle dlya ellipticheskogo uravneniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:1 (2015), 19–43
A. K. Gushchin, “Lp-estimates for solutions of second-order elliptic equation Dirichlet problem”, Theoret. and Math. Phys., 174:2 (2013), 209–219
A. K. Gushchin, “The Dirichlet problem for a second-order elliptic equation with an Lp boundary function”, Sb. Math., 203:1 (2012), 1–27
A. K. Guschin, “Otsenki resheniya zadachi Dirikhle s granichnoi funktsiei iz Lp”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(22) (2011), 53–67
Gushchin A.K., “Solvability of the Dirichlet problem for a second-order elliptic equation with a boundary function from Lp”, Dokl. Math., 83:2 (2011), 219–221
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