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Teoreticheskaya i Matematicheskaya Fizika, 2008, Volume 157, Number 3, Pages 345–363
DOI: https://doi.org/10.4213/tmf6284
(Mi tmf6284)
 

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

A strengthening of the interior Hölder continuity property for solutions of the Dirichlet problem for a second-order elliptic equation

A. K. Gushchin

Steklov Mathematical Institute, Russian Academy of Sciences
References:
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.
Received: 03.04.2008
English version:
Theoretical and Mathematical Physics, 2008, Volume 157, Issue 3, Pages 1655–1670
DOI: https://doi.org/10.1007/s11232-008-0138-0
Bibliographic databases:
Document Type: Article
Language: Russian
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
Citation in format AMSBIB
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  • https://doi.org/10.4213/tmf6284
  • https://www.mathnet.ru/eng/tmf/v157/i3/p345
  • This publication is cited in the following 12 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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