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Izv. Akad. Nauk SSSR Ser. Mat., 1982, Volume 46, Issue 3, Pages 487–523 (Mi izv1637)  

This article is cited in 16 scientific papers (total in 16 papers)

Boundedly nonhomogeneous elliptic and parabolic equations

N. V. Krylov

Abstract: This paper considers elliptic equations of the form
\begin{equation*} 0=F(u_{x^ix^j},u_{x^i},u,1,x) \tag{$*$} \end{equation*}
and parabolic equations of the form
\begin{equation*} u_t=F(u_{x^ix^j},u_{x^i},u,1,t,x), \tag{$**$} \end{equation*}
where $F(u_{ij},u_i,u,\beta,x)$ and $F(u_{ij},u_i,u,\beta,t,x)$ are positive homogeneous functions of the first order of homogeneity with respect to $(u_{ij},u_i,u,\beta)$, convex upwards with respect $u_{ij}$ and satisfying a uniform condition of strict ellipticity. Under certain smoothness conditions on $F$ and boundedness from above of the second derivatives of $F$ with respect to $(u_{ij},u_i,u)$, solvability is established for these equations of a problem over the whole space, of the Dirichlet problem in a domain with a sufficiently regular boundary (for the equation ($*$)), and of the Cauchy problem and the first boundary value problem (for equation ($**$)). Solutions are sought in the classes $C^{2+\alpha}$, and their existence is proved with the aid of internal a priori estimates in $C^{2+\alpha}$.
Bibliography: 29 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1983, 20:3, 459–492

Bibliographic databases:

UDC: 517.9
MSC: 35A05, 35J15, 35K10
Received: 09.07.1981

Citation: N. V. Krylov, “Boundedly nonhomogeneous elliptic and parabolic equations”, Izv. Akad. Nauk SSSR Ser. Mat., 46:3 (1982), 487–523; Math. USSR-Izv., 20:3 (1983), 459–492

Citation in format AMSBIB
\by N.~V.~Krylov
\paper Boundedly nonhomogeneous elliptic and parabolic equations
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1982
\vol 46
\issue 3
\pages 487--523
\jour Math. USSR-Izv.
\yr 1983
\vol 20
\issue 3
\pages 459--492

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    This publication is cited in the following articles:
    1. N. V. Krylov, “Boundedly nonhomogeneous elliptic and parabolic equations in a domain”, Math. USSR-Izv., 22:1 (1984), 67–97  mathnet  crossref  mathscinet  zmath
    2. L. Caffarelli, L. Nirenberg, J. Spruck, “The Dirichlet problem for nonlinear second-order elliptic equations I. Monge-ampégre equation”, Comm Pure Appl Math, 37:3 (1984), 369  crossref  mathscinet  zmath
    3. L. Caffarelli, J. J. Kohn, L. Nirenberg, J. Spruck, “The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex monge-ampère, and uniformaly elliptic, equations”, Comm Pure Appl Math, 38:2 (1985), 209  crossref  mathscinet  zmath
    4. N. M. Ivochkina, “Solution of the Dirichlet problem for some equations of Monge–Aampére type”, Math. USSR-Sb., 56:2 (1987), 403–415  mathnet  crossref  mathscinet  zmath
    5. S. V. Anulova, “An estimate of the probability that a degenerate diffusion process hits a set of positive measure”, Math. USSR-Izv., 28:2 (1987), 201–231  mathnet  crossref  mathscinet  zmath
    6. M. V. Safonov, “On the classical solution of nonlinear elliptic equations of second order”, Math. USSR-Izv., 33:3 (1989), 597–612  mathnet  crossref  mathscinet  zmath
    7. Kazuo Amano, “The Dirichlet problem for degenerate elliptic 2-dimensional Monge-Ampère equation”, BAZ, 37:3 (1988), 389  crossref
    8. Sun-Yung Alice Chang, Paul C. Yang, “The inequality of Moser and Trudinger and applications to conformal geometry”, Comm Pure Appl Math, 56:8 (2003), 1135  crossref  mathscinet  zmath
    9. Mariko Arisawa, “Long time averaged reflection force and homogenization of oscillating Neumann boundary conditions”, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 20:2 (2003), 293  crossref
    10. Herbert Koch, Giovanni Leoni, Massimiliano Morini, “On optimal regularity of free boundary problems and a conjecture of De Giorgi”, Comm Pure Appl Math, 58:8 (2005), 1051  crossref  mathscinet  zmath
    11. Luis A. Caffarelli, Panagiotis E. Souganidis, “A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs”, Comm Pure Appl Math, 61:1 (2008), 1  crossref  mathscinet  zmath
    12. H. Kim, M. Safonov, “Boundary Harnack principle for second order elliptic equations with unbounded drift”, J Math Sci, 2011  crossref
    13. Damião.J.. Araújo, Gleydson Ricarte, E.V.. Teixeira, “Geometric gradient estimates for solutions to degenerate elliptic equations”, Calc. Var, 2014  crossref
    14. Ben Andrews, Mat Langford, “Cylindrical estimates for hypersurfaces moving by convex curvature functions”, Anal. PDE, 7:5 (2014), 1091  crossref
    15. Valentino Tosatti, Yu Wang, Ben Weinkove, Xiaokui Yang, “
      $$C^{2,\alpha }$$
      C 2 , α estimates for nonlinear elliptic equations in complex and almost complex geometry”, Calc. Var, 2014  crossref
    16. M.N.. Ivaki, “Convex bodies with pinched Mahler volume under the centro-affine normal flows”, Calc. Var, 2014  crossref
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