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Sibirsk. Mat. Zh., 2005, Volume 46, Number 5, Pages 1036–1052 (Mi smj1020)  

This article is cited in 15 scientific papers (total in 15 papers)

On the interior smoothness of solutions to second-order elliptic equations

A. K. Gushchin

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We study the interior smoothness properties of solutions to a linear second-order uniformly elliptic equation in selfadjoint form without lower-order terms and with measurable bounded coefficients. In terms of membership in a special function space we combine and supplement some properties of solutions such as membership in the Sobolev space $W^1_{2,\mathrm{loc}}$ and Holder continuity. We show that the membership of solutions in the introduced space which we establish in this article gives some new properties that do not follow from Holder continuity and the membership in $W^1_{2,\mathrm{loc}}$.

Keywords: elliptic equation, function spaces, smoothness of solutions

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English version:
Siberian Mathematical Journal, 2005, 46:5, 826–840

Bibliographic databases:

UDC: 517.9
Received: 15.04.2005

Citation: A. K. Gushchin, “On the interior smoothness of solutions to second-order elliptic equations”, Sibirsk. Mat. Zh., 46:5 (2005), 1036–1052; Siberian Math. J., 46:5 (2005), 826–840

Citation in format AMSBIB
\by A.~K.~Gushchin
\paper On the interior smoothness of solutions to second-order elliptic equations
\jour Sibirsk. Mat. Zh.
\yr 2005
\vol 46
\issue 5
\pages 1036--1052
\jour Siberian Math. J.
\yr 2005
\vol 46
\issue 5
\pages 826--840

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    This publication is cited in the following articles:
    1. Gushchin AK, “Smoothness of solutions to the Dirichlet problem for a second-order elliptic equation with a square integrable boundary function”, Doklady Mathematics, 76:1 (2007), 486–489  crossref  mathscinet  zmath  isi  elib  scopus
    2. L. M. Kozhevnikova, “Behaviour at infinity of solutions of pseudodifferential elliptic equations in unbounded domains”, Sb. Math., 199:8 (2008), 1169–1200  mathnet  crossref  crossref  mathscinet  isi  elib
    3. A. K. Gushchin, “A strengthening of the interior Hölder continuity property for solutions of the Dirichlet problem for a second-order elliptic equation”, Theoret. and Math. Phys., 157:3 (2008), 1655–1670  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. A. R. Gerfanov, F. Kh. Mukminov, “Shirokii klass edinstvennosti resheniya dlya neravnomerno ellipticheskogo uravneniya v neogranichennoi oblasti”, Ufimsk. matem. zhurn., 1:3 (2009), 11–27  mathnet  zmath  elib
    5. V. F. Gilimshina, F. Kh. Mukminov, “On the decay of solutions of non-uniformly elliptic equations”, Izv. Math., 75:1 (2011), 53–71  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. 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
    7. Gushchin A.K., “Solvability of the Dirichlet problem for a second-order elliptic equation with a boundary function from $L_{p}$”, Doklady Mathematics, 83:2 (2011), 219–221  crossref  mathscinet  zmath  zmath  isi  elib  elib  scopus
    8. 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
    9. 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
    10. 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
    11. 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
    12. A. K. Gushchin, “V.A. Steklov's work on equations of mathematical physics and development of his results in this field”, Proc. Steklov Inst. Math., 289 (2015), 134–151  mathnet  crossref  crossref  isi  elib
    13. 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
    14. 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
    15. 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
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