## The Dirichlet problem for nonlinear second-order elliptic equations. II: Complex Monge-Ampère, and uniformly elliptic, equations.(English)Zbl 0598.35048

[For Part I, see ibid. 37, 369-402 (1984; Zbl 0598.35047).]
This is the second paper in a series of three papers devoted to the Dirichlet problem for second-order nonlinear elliptic equations. The third one will appear in Acta Mathematica. In this paper the authors treat the problem $F(x,u,Du,D^ 2u)=0\quad in\quad \Omega,\quad u=\phi \quad on\quad \partial \Omega.$ The function F is smooth for $$x\in {\bar \Omega}$$ in all the arguments, $\sum (\partial F/\partial u_{ij})\xi_ i\xi_ j>0\quad for\quad \xi =(\xi_ 1,..,\xi_ n)\neq 0,$ and F is a concave function of the second derivatives $$\{u_{ij}\}$$. (1) The paper contains three sections. In the first section the $$C^ 2$$ a priori estimates for elliptic complex Monge- Ampère equations are derived. The principal contribution of the second section is the derivation of a logarithmic modulus of continuity of $$u_{ij}$$ near the boundary. The last section is a self-contained treatment of a rather general class of ”uniformly elliptic” operators satisfying (1). It is worth mentioning that this paper is in close relation to works of N. V. Krylov and N. S. Trudinger which are stated in the references.
Reviewer: P.Drabek

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35B50 Maximum principles in context of PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B45 A priori estimates in context of PDEs

Zbl 0598.35047
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### References:

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