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 Izv. Akad. Nauk SSSR Ser. Mat., 1988, Volume 52, Issue 6, Pages 1272–1287 (Mi izv1230)

On the classical solution of nonlinear elliptic equations of second order

M. V. Safonov

Abstract: The Dirichlet problem $E(u_{x_ix_j},u_{x_i},u,x)=0$ in $\Omega\subset R^d$, $u=\varphi$ on $\partial\Omega$, is considered for nonlinear elliptic equations, including Bellman equations with “coefficients” in the Hölder space $C^{\alpha}(\overline\Omega)$. It is proved that if $\alpha>0$ is sufficiently small, then this problem is solvable in $C^{2+\alpha}_{\mathrm{loc}}(\Omega)\cap C(\overline\Omega)$. If in addition $\partial\Omega\in C^{2+\alpha}$ and $\varphi\in C^{2+\alpha}(\overline\Omega)$, then the solution belongs to $C^{2+\alpha}(\overline\Omega)$.
Bibliography: 18 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1989, 33:3, 597–612

Bibliographic databases:

UDC: 517.957
MSC: 35J65

Citation: M. V. Safonov, “On the classical solution of nonlinear elliptic equations of second order”, Izv. Akad. Nauk SSSR Ser. Mat., 52:6 (1988), 1272–1287; Math. USSR-Izv., 33:3 (1989), 597–612

Citation in format AMSBIB
\Bibitem{Saf88}
\by M.~V.~Safonov
\paper On~the classical solution of nonlinear elliptic equations of second order
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1988
\vol 52
\issue 6
\pages 1272--1287
\mathnet{http://mi.mathnet.ru/izv1230}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=984219}
\zmath{https://zbmath.org/?q=an:0682.35048}
\transl
\jour Math. USSR-Izv.
\yr 1989
\vol 33
\issue 3
\pages 597--612
\crossref{https://doi.org/10.1070/IM1989v033n03ABEH000858}

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This publication is cited in the following articles:
1. Kovats Jay, “Fully nonlinear elliptic equations and the dini condition”, Communications in Partial Differential Equations, 22:11-12 (1997), 1911
2. M. G Crandall, M Kocan, A. Świech, “Lp- Theory for fully nonlinear uniformly parabolic equations”, Communications in Partial Differential Equations, 25:11-12 (2000), 1997
3. O ALVAREZ, M BARDI, C MARCHI, “Multiscale problems and homogenization for second-order Hamilton–Jacobi equations”, Journal of Differential Equations, 243:2 (2007), 349
4. Orazio Arena, Pasquale Buonocore, “On a variational problem for radial solutions to extremal elliptic equations”, Annali di Matematica, 2008
5. Fabio Camilli, Claudio Marchi, “Rates of convergence in periodic homogenization of fully nonlinear uniformly elliptic PDEs”, Nonlinearity, 22:6 (2009), 1481
6. Jiakun Liu, Neil S. Trudinger, Xu-Jia Wang, “InteriorC2,αRegularity for Potential Functions in Optimal Transportation”, Communications in Partial Differential Equations, 35:1 (2009), 165
7. Claudio Marchi, Fabio Camilli, “On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems”, NHM, 6:1 (2011), 61
8. N. V. Krylov, “On C 1+α regularity of solutions of Isaacs parabolic equations with VMO coefficients”, Nonlinear Differ. Equ. Appl, 2013
9. Claudio Marchi, “Continuous dependence estimates for the ergodic problem of Bellman equation with an application to the rate of convergence for the homogenization problem”, Calc. Var, 2013
10. Bruno Strulovici, Martin Szydlowski, “On the smoothness of value functions and the existence of optimal strategies in diffusion models”, Journal of Economic Theory, 2015
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