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Izv. Akad. Nauk SSSR Ser. Mat., 1978, Volume 42, Issue 5, Pages 1050–1062 (Mi izv1926)  

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

On the maximum principle for nonlinear parabolic and elliptic equations

N. V. Krylov


Abstract: A maximum principle in Sobolev spaces is proved for nonlinear elliptic and parabolic equations. The proof is based on estimates for the maximum of the solutions of a parabolic equation with measurable coefficients, in terms of the $\mathscr L_p$ norm of the right side. The results of the paper are analogous to results of A. D. Aleksandrov on elliptic equations.
Bibliography: 10 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1979, 13:2, 335–347

Bibliographic databases:

UDC: 517.9
MSC: Primary 35K55; Secondary 35B45, 35J60
Received: 11.05.1977

Citation: N. V. Krylov, “On the maximum principle for nonlinear parabolic and elliptic equations”, Izv. Akad. Nauk SSSR Ser. Mat., 42:5 (1978), 1050–1062; Math. USSR-Izv., 13:2 (1979), 335–347

Citation in format AMSBIB
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\by N.~V.~Krylov
\paper On the maximum principle for nonlinear parabolic and elliptic equations
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1978
\vol 42
\issue 5
\pages 1050--1062
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=513913}
\zmath{https://zbmath.org/?q=an:0421.35041|0398.35008}
\transl
\jour Math. USSR-Izv.
\yr 1979
\vol 13
\issue 2
\pages 335--347
\crossref{https://doi.org/10.1070/IM1979v013n02ABEH002046}
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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. N. V. Krylov, “On the limit passage in parabolic Bellman equations”, Math. USSR-Izv., 13:3 (1979), 677–684  mathnet  crossref  mathscinet  zmath  isi
    2. N. V. Krylov, M. V. Safonov, “A certain property of solutions of parabolic equations with measurable coefficients”, Math. USSR-Izv., 16:1 (1981), 151–164  mathnet  crossref  mathscinet  zmath  isi
    3. V. V. Zhikov, M. M. Sirazhudinov, “On $G$-compactness of a class of nondivergence elliptic operators of second order”, Math. USSR-Izv., 19:1 (1982), 27–40  mathnet  crossref  mathscinet  zmath
    4. Jan Chabrowski, Rudolf Výborný, “Maximum principle for non-linear degenerate equations of the parabolic type”, BAZ, 25:2 (1982), 251  crossref
    5. O. A. Ladyzhenskaya, N. N. Ural'tseva, “A survey of results on the solubility of boundary-value problems for second-order uniformly elliptic and parabolic quasi-linear equations having unbounded singularities”, Russian Math. Surveys, 41:5 (1986), 1–31  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. N. V. Krylov, “On estimates of the maximum of a solution of a parabolic equation and estimates of the distribution of a semimartingale”, Math. USSR-Sb., 58:1 (1987), 207–221  mathnet  crossref  mathscinet  zmath
    7. Yu. A. Alkhutov, I. T. Mamedov, “The first boundary value problem for nondivergence second order parabolic equations with discontinuous coefficients”, Math. USSR-Sb., 59:2 (1988), 471–495  mathnet  crossref  mathscinet  zmath
    8. R. Výborný, “The Hadamard three-circles theorems for nonlinear equations”, J Austral Math Soc, 49:2 (1990), 297  crossref
    9. N. V. Krylov, “On C 1+α regularity of solutions of Isaacs parabolic equations with VMO coefficients”, Nonlinear Differ. Equ. Appl, 2013  crossref
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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