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Mat. Sb. (N.S.), 1986, Volume 131(173), Number 1(9), Pages 113–125 (Mi msb1908)  

This article is cited in 20 scientific papers (total in 22 papers)

On the best Hölder exponents for generalized solutions of the Dirichlet problem for a second order elliptic equation

V. A. Kondrat'ev, J. Kopáček, O. A. Oleinik


Abstract: The authors study the behavior of generalized solutions of the Dirichlet problem for a second order elliptic equation in a neighborhood of a boundary point. Under certain assumptions on the structure of the boundary of the domain in such a neighborhood, and on the coefficients of the equation, a power modulus of continuity is obtained at the boundary point for generalized solutions of the Dirichlet problem, the exponent being best possible for domains with the indicated structure of the boundary near that point. The assumptions on the coefficients of the equation are essential, as an example shows. With the help of the indicated results on the modulus of continuity at boundary points, it is then shown that generalized solutions belong to Hölder spaces in the closure of the domain, the Hölder exponent again being best possible for the class of domains under consideration.
Bibliography: 8 titles.

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English version:
Mathematics of the USSR-Sbornik, 1988, 59:1, 113–127

Bibliographic databases:

UDC: 517.95
MSC: 35J25, 35D10
Received: 20.11.1985

Citation: V. A. Kondrat'ev, J. Kopáček, O. A. Oleinik, “On the best Hölder exponents for generalized solutions of the Dirichlet problem for a second order elliptic equation”, Mat. Sb. (N.S.), 131(173):1(9) (1986), 113–125; Math. USSR-Sb., 59:1 (1988), 113–127

Citation in format AMSBIB
\Bibitem{KonKopOle86}
\by V.~A.~Kondrat'ev, J.~Kop\'a{\v{c}}ek, O.~A.~Oleinik
\paper On the best H\"older exponents for generalized solutions of the Dirichlet problem for a~second order elliptic equation
\jour Mat. Sb. (N.S.)
\yr 1986
\vol 131(173)
\issue 1(9)
\pages 113--125
\mathnet{http://mi.mathnet.ru/msb1908}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=868604}
\zmath{https://zbmath.org/?q=an:0633.35020}
\transl
\jour Math. USSR-Sb.
\yr 1988
\vol 59
\issue 1
\pages 113--127
\crossref{https://doi.org/10.1070/SM1988v059n01ABEH003127}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. V. Chistyakov, “On some qualitative properties of the solutions of a non-divergent semilinear second-order parabolic equation”, Russian Math. Surveys, 41:5 (1986), 133–134  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. A. K. Gushchin, “On the Dirichlet problem for a second-order elliptic equation”, Math. USSR-Sb., 65:1 (1990), 19–66  mathnet  crossref  mathscinet  zmath
    3. Catherine Bandle, Matts Essén, “On positive solutions of Emden equations in cone-like domains”, Arch Rational Mech Anal, 112:4 (1990), 319  crossref  mathscinet  zmath
    4. M. V. Borsuk, “Best-possible estimates of solutions of the Dirichlet problem for linear elliptic nondivergence equations of second order in a neighborhood of a conical point of the boundary”, Math. USSR-Sb., 74:1 (1993), 185–201  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. Azzam A., Kondratev V., “Schauder Type Estimates of Solutions of 2nd-Order Elliptic-Systems in Divergence Form, in Nonregular Domains”, Commun. Partial Differ. Equ., 16:12 (1991), 1857–1878  crossref  mathscinet  zmath  isi
    6. Borsuk M., “Non-Improvable Estimates for Solutions of Dirichlet Problem for 2nd-Order Linear Elliptic Nondivergent Equations in Domain with Corners”, 317, no. 4, 1991, 790–792  mathscinet  zmath  isi
    7. Azzam A., “Asymptotic Expansions of Solutions of the 1st Boundary-Value Problem for Elliptic-Equations Near Corners”, Math. Nachr., 153 (1991), 9–31  crossref  mathscinet  zmath  isi
    8. Kondratev V., “On Solutions of Weakly Nonlinear Elliptic-Equations in the Neighborhood of a Conic Point at the Boundary”, Differ. Equ., 29:2 (1993), 246–252  mathnet  mathscinet  zmath  isi
    9. A. K. Gushchin, V. P. Mikhailov, “On solvability of nonlocal problems for a second-order elliptic equation”, Russian Acad. Sci. Sb. Math., 81:1 (1995), 101–136  mathnet  crossref  mathscinet  zmath  isi
    10. N. S. Bakhvalov, S. P. Novikov, A. T. Fomenko, “Ol'ga Arsen'evna Oleinik (on her seventieth birthday)”, Russian Math. Surveys, 50:4 (1995), 837–848  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    11. Yu. A. Alkhutov, “$L_p$-estimates of the solution of the Dirichlet problem for second-order elliptic equations”, Sb. Math., 189:1 (1998), 1–17  mathnet  crossref  crossref  mathscinet  zmath  isi
    12. Kondrat'ev V., “Completeness of the Systems of Root Functions of Elliptic Operators in Banach Spaces”, Russ. J. Math. Phys., 6:2 (1999), 194–201  mathscinet  zmath  isi
    13. Kondrat'ev V., Nikishkin V., “On the Behavior of Solutions of Elliptic Equations in a Neighborhood of a Crack with Nonsmooth Front”, Russ. J. Math. Phys., 9:1 (2002), 106–111  mathscinet  zmath  isi
    14. M. V. Borsuk, “Second-order degenerate elliptic boundary value problems in nonsmooth domains”, Journal of Mathematical Sciences, 146:5 (2007), 6071–6212  mathnet  crossref  mathscinet  zmath  elib
    15. M. S. Agranovich, I. V. Astashova, L. A. Bagirov, V. V. Vlasov, V. V. Zhikov, Yu. S. Ilyashenko, V. V. Kozlov, A. A. Kon'kov, S. I. Pokhozhaev, E. V. Radkevich, N. Kh. Rozov, I. N. Sergeev, A. L. Skubachevskii, G. A. Chechkin, A. S. Shamaev, T. A. Shaposhnikova, “Vladimir Alexandrovich Kondratiev. July 2, 1935 – March 11, 2010”, Journal of Mathematical Sciences, 190:1 (2013), 1–7  mathnet  crossref  mathscinet
    16. Gushchin A.K., “Estimates of the Nontangential Maximal Function for Solutions of a Second-Order Elliptic Equation”, Dokl. Math., 86:2 (2012), 667–669  mathnet  crossref  mathscinet  zmath  isi  elib
    17. 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
    18. A. K. Gushchin, “$L_p$-estimates for the nontangential maximal function of the solution to a second-order elliptic equation”, Sb. Math., 207:10 (2016), 1384–1409  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    19. A. K. Gushchin, “The Luzin area integral and the nontangential maximal function for solutions to a second-order elliptic equation”, Sb. Math., 209:6 (2018), 823–839  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    20. 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
    21. A. K. Gushchin, “The boundary values of solutions of an elliptic equation”, Sb. Math., 210:12 (2019), 1724–1752  mathnet  crossref  crossref  mathscinet  isi
    22. 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
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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