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TMF, 2008, Volume 157, Number 3, Pages 345–363 (Mi tmf6284)  

This article is cited in 9 scientific papers (total in 9 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

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

DOI: https://doi.org/10.4213/tmf6284

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English version:
Theoretical and Mathematical Physics, 2008, 157:3, 1655–1670

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Received: 03.04.2008

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
\by A.~K.~Gushchin
\paper A~ strengthening of the~interior H\"older continuity property for solutions of the~Dirichlet problem for a~second-order elliptic equation
\jour TMF
\yr 2008
\vol 157
\issue 3
\pages 345--363
\jour Theoret. and Math. Phys.
\yr 2008
\vol 157
\issue 3
\pages 1655--1670

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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. A. K. Guschin, “Otsenki resheniya zadachi Dirikhle s granichnoi funktsiei iz $L_p$”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(22) (2011), 53–67  mathnet  crossref  elib
    2. Gushchin A.K., “Solvability of the Dirichlet problem for a second-order elliptic equation with a boundary function from $L_p$”, Dokl. Math., 83:2 (2011), 219–221  crossref  mathscinet  zmath  isi  elib  elib  scopus
    3. A. K. Gushchin, “The Dirichlet problem for a second-order elliptic equation with an $L_p$ boundary function”, Sb. Math., 203:1 (2012), 1–27  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. A. K. Gushchin, “$L_p$-estimates for solutions of second-order elliptic equation Dirichlet problem”, Theoret. and Math. Phys., 174:2 (2013), 209–219  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. A. K. Guschin, “$L_p$-otsenki nekasatelnoi maksimalnoi funktsii dlya reshenii ellipticheskogo uravneniya vtorogo poryadka”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 53–69  mathnet  crossref
    6. A. K. Guschin, “O zadache Dirikhle dlya ellipticheskogo uravneniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:1 (2015), 19–43  mathnet  crossref  zmath  elib
    7. A. K. Gushchin, “A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation”, Proc. Steklov Inst. Math., 301 (2018), 44–64  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    8. A. K. Gushchin, “The boundary values of solutions of an elliptic equation”, Sb. Math., 210:12 (2019), 1724–1752  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    9. A. K. Gushchin, “On the Existence of $L_2$ Boundary Values of Solutions to an Elliptic Equation”, Proc. Steklov Inst. Math., 306 (2019), 47–65  mathnet  crossref  crossref  mathscinet  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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