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Izv. Akad. Nauk SSSR Ser. Mat., 1983, Volume 47, Issue 1, Pages 75–108 (Mi izv1382)  

This article is cited in 35 scientific papers (total in 36 papers)

Boundedly nonhomogeneous elliptic and parabolic equations in a domain

N. V. Krylov


Abstract: In this paper the Dirichlet problem is studied for equations of the form $0=F(u_{x^ix^j},u_{x^i},u,1,x)$ and also the first boundary value problem for equations of the form $u_t=F(u_{x^ix^j},u_{x^i},u,1,t,x)$, 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 degree in $(u_{ij},u_i,u,\beta)$, convex upwards in $(u_{ij})$, that satisfy a uniform strict ellipticity condition. Under certain smoothness conditions on $F$ and when the second derivatives of $F$ with respect to $(u_{ij},u_i,u,x)$ are bounded above, the $C^{2+\alpha}$ solvability of these problems in smooth domains is proved. In the course of the proof, a priori estimates in $C^{2+\alpha}$ on the boundary are constructed, and convexity and restrictions on the second derivatives of $F$ are not used in the derivation.
Bibliography: 13 titles.

Full text: PDF file (3054 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1984, 22:1, 67–97

Bibliographic databases:

UDC: 517.9
MSC: Primary 35A05, 35B45, 35J25, 35K20; Secondary 26B35, 35B65, 35J60, 35K55
Received: 30.11.1981

Citation: N. V. Krylov, “Boundedly nonhomogeneous elliptic and parabolic equations in a domain”, Izv. Akad. Nauk SSSR Ser. Mat., 47:1 (1983), 75–108; Math. USSR-Izv., 22:1 (1984), 67–97

Citation in format AMSBIB
\Bibitem{Kry83}
\by N.~V.~Krylov
\paper Boundedly nonhomogeneous elliptic and parabolic equations in a~domain
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1983
\vol 47
\issue 1
\pages 75--108
\mathnet{http://mi.mathnet.ru/izv1382}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=688919}
\zmath{https://zbmath.org/?q=an:0578.35024}
\transl
\jour Math. USSR-Izv.
\yr 1984
\vol 22
\issue 1
\pages 67--97
\crossref{https://doi.org/10.1070/IM1984v022n01ABEH001434}


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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 degenerate nonlinear elliptic equations. II”, Math. USSR-Sb., 49:1 (1984), 207–228  mathnet  crossref  mathscinet  zmath
    2. N. V. Krylov, “On degenerate nonlinear elliptic equations”, Math. USSR-Sb., 48:2 (1984), 307–326  mathnet  crossref  mathscinet  zmath
    3. 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
    4. 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
    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. 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. Chin-Yuan Lin, “Fully nonlinear parabolic boundary value problems in higher space dimensions (II)”, Nonlinear Analysis: Theory, Methods & Applications, 15:4 (1990), 355  crossref
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    10. C.M.. Corona, “Monge-Ampere Equations on Convex Regions of The Plane”, Communications in Partial Differential Equations, 16:1 (1991), 43  crossref
    11. Gary M. Lieberman, “Boundary and initial regularity for solutions of degenerate parabolic equations”, Nonlinear Analysis: Theory, Methods & Applications, 20:5 (1993), 551  crossref
    12. Guan B., Spruck J., “Boundary-Value-Problems on S(N) for Surfaces of Constant Gauss Curvature”, Ann. Math., 138:3 (1993), 601–624  crossref  mathscinet  zmath  isi
    13. Rosenberg H., Spruck J., “On the Existence of Convex Hypersurfaces of Constant Gauss Curvature in Hyperbolic Space”, J. Differ. Geom., 40:2 (1994), 379–409  mathscinet  zmath  isi
    14. Amel Atllah, “Problème de Dirichlet pour des équations de monge-ampère réelles relatives à des mètriques Riemanniennes par”, Communications in Partial Differential Equations, 21:1-2 (1996), 35  crossref
    15. Kovats Jay, “Fully nonlinear elliptic equations and the dini condition”, Communications in Partial Differential Equations, 22:11-12 (1997), 1911  crossref
    16. 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
    17. Lopez R., “Some a Priori Bounds for Solutions of the Constant Gauss Curvature Equation”, J. Differ. Equ., 194:1 (2003), 185–197  crossref  mathscinet  zmath  isi
    18. Bo Guan, Joel Spruck, “Locally convex hypersurfaces of constant curvature with boundary”, Comm Pure Appl Math, 57:10 (2004), 1311  crossref  mathscinet  zmath  elib
    19. 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
    20. A. I. Nazarov, “Hölder estimates for solutions to degenerate nondivergence elliptic and parabolic equations”, St. Petersburg Math. J., 21:4 (2010), 635–650  mathnet  crossref  mathscinet  zmath  isi
    21. H. Kim, M. Safonov, “Boundary Harnack principle for second order elliptic equations with unbounded drift”, J Math Sci, 2011  crossref
    22. N. M. Ivochkina, S. I. Prokof’eva, G. V. Yakunina, “The Gårding cones in the modern theory of fully nonlinear second order differential equations”, J Math Sci, 184:3 (2012), 295  crossref
    23. Xiuxiong Chen, Weiyong He, “The complex Monge–Ampère equation on compact Kähler manifolds”, Math. Ann, 354:4 (2012), 1583  crossref
    24. N. M. Ivochkina, “From Gårding's cones to $p$-convex hypersurfaces”, Journal of Mathematical Sciences, 201:5 (2014), 634–644  mathnet  crossref  mathscinet
    25. A. Sadullaev, B. Abdullaev, “Potential theory in the class of $m$-subharmonic functions”, Proc. Steklov Inst. Math., 279 (2012), 155–180  mathnet  crossref  mathscinet  isi  elib
    26. Yongpan Huang, Dongsheng Li, Lihe Wang, “Lateral boundary differentiability of solutions of parabolic equations on cylindrical convex domains”, Journal of Mathematical Analysis and Applications, 2013  crossref
    27. Feida Jiang, Neil S. Trudinger, Xiao-Ping Yang, “On the Dirichlet problem for Monge-Ampère type equations”, Calc. Var, 2013  crossref
    28. Yongpan Huang, Dongsheng Li, Lihe Wang, “Boundary behavior of solutions of elliptic equations in nondivergence form”, manuscripta math, 2013  crossref
    29. E.V.. Teixeira, “Universal Moduli of Continuity for Solutions to Fully Nonlinear Elliptic Equations”, Arch Rational Mech Anal, 2013  crossref
    30. Yu. A. Alkhutov, “Hölder continuity of solutions of nondivergent degenerate second-order elliptic equations”, J. Math. Sci. (N. Y.), 197:2 (2014), 151–174  mathnet  crossref  elib
    31. Damião.J.. Araújo, Gleydson Ricarte, E.V.. Teixeira, “Geometric gradient estimates for solutions to degenerate elliptic equations”, Calc. Var, 2014  crossref
    32. N. M. Ivochkina, N. V. Filimonenkova, “On new structures in the theory of fully nonlinear equations”, Journal of Mathematical Sciences, 233:4 (2018), 480–494  mathnet  crossref
    33. N. M. Ivochkina, N. V. Filimonenkova, “Konusy Gordinga i uravneniya Bellmana v teorii gessianovskikh operatorov i uravnenii”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 63, no. 4, Rossiiskii universitet druzhby narodov, M., 2017, 615–626  mathnet  crossref
    34. A. A. Borisenko, A. Yu. Vesnin, N. M. Ivochkina, “On the 100th anniversary of the birth of Aleksei Vasil'evich Pogorelov”, Russian Math. Surveys, 74:6 (2019), 1135–1157  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    35. N. V. Filimonenkova, “Geometriya $m$-gessianovskikh uravnenii”, Materialy mezhdunarodnoi konferentsii “Geometricheskie metody v teorii upravleniya i matematicheskoi fizike”, posvyaschennoi 70-letiyu S.L. Atanasyana, 70-letiyu I.S. Krasilschika, 70-letiyu A.V. Samokhina, 80-letiyu V.T. Fomenko. Ryazanskii gosudarstvennyi universitet im. S.A. Esenina, Ryazan, 25–28 sentyabrya 2018 g. Chast 2, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 169, VINITI RAN, M., 2019, 98–115  mathnet  crossref
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