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Some estimates for classical solutions of fully nonlinear second order elliptic equations. (Chinese. English summary) Zbl 0656.35041

The paper deals with the classical solutions of the Dirichlet problem of fully nonlinear second order elliptic equation \[ F(D^ 2u,Du,u,x)=0\quad in\quad \Omega;\quad u=0\quad on\quad \partial \Omega \] where \(\Omega\) is a bounded domain in \(R^ n\). Besides some structure conditions the function F(r,p,z,x) is assumed to be only once differentiable (or to be Hölder continuous in some cases) with respect to x. So the conditions on the differentiability of F with respect to x demanded by L. C. Evans [Commun. Pure Appl. Math. 35, 333-363 (1982; Zbl 0469.35022)] and N. S. Trudinger [Trans. Am. Math. Soc. 278, 751-769 (1983; Zbl 0518.35036)] are weakened. Using freezing coefficients and quasilinearization and a new result of M. V. Safonov [Sov. Math., Dokl. 30, 482-485 (1984); translation from Dokl. Akad. Nauk SSSR 278, 810-813 (1984; Zbl 0595.35011)] about Bellman equations, \(C^{2+\alpha}\) estimates are obtained. Some estimates similar to linear equations are also obtained for special cases.
Reviewer: J.H.Tian

MSC:

35J60 Nonlinear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35B45 A priori estimates in context of PDEs
35J15 Second-order elliptic equations
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