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

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

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
\Bibitem{Kry82} \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 \mathnet{http://mi.mathnet.ru/izv1637} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=661144} \zmath{https://zbmath.org/?q=an:0529.35026|0511.35002} \transl \jour Math. USSR-Izv. \yr 1983 \vol 20 \issue 3 \pages 459--492 \crossref{https://doi.org/10.1070/IM1983v020n03ABEH001360} 

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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. N. V. Krylov, “Boundedly nonhomogeneous elliptic and parabolic equations in a domain”, Math. USSR-Izv., 22:1 (1984), 67–97
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
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
4. N. M. Ivochkina, “Solution of the Dirichlet problem for some equations of Monge–Aampére type”, Math. USSR-Sb., 56:2 (1987), 403–415
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
6. M. V. Safonov, “On the classical solution of nonlinear elliptic equations of second order”, Math. USSR-Izv., 33:3 (1989), 597–612
7. Kazuo Amano, “The Dirichlet problem for degenerate elliptic 2-dimensional Monge-Ampère equation”, BAZ, 37:3 (1988), 389
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
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
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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
12. H. Kim, M. Safonov, “Boundary Harnack principle for second order elliptic equations with unbounded drift”, J Math Sci, 2011
13. Damião.J.. Araújo, Gleydson Ricarte, E.V.. Teixeira, “Geometric gradient estimates for solutions to degenerate elliptic equations”, Calc. Var, 2014
14. Ben Andrews, Mat Langford, “Cylindrical estimates for hypersurfaces moving by convex curvature functions”, Anal. PDE, 7:5 (2014), 1091
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
16. M.N.. Ivaki, “Convex bodies with pinched Mahler volume under the centro-affine normal flows”, Calc. Var, 2014
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