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A weak Bernstein method for fully non-linear elliptic equations. (English) Zbl 0733.35014
The present paper is devoted to gradient bounds for solutions of general elliptic equations of the type \(F(x,u,Du,D^ 2u)=0\) in \(\Omega\), for \(F\) continuous from \(\Omega \times\mathbb R\times\mathbb R^ n\times S^ n\) into \(\mathbb R\), \(S^ n\) the space of symmetric \(n\times n\) matrices with the usual partial ordering. The idea is to prove results of Bernstein’s type \[ \sup_{\Omega} | Du|^ 2\leq \max \quad (\text{constant},\sup_{\partial \Omega}| Du|^ 2), \] for viscosity solutions of the equation above, that is, without differentiating the equation, and by assuming the viscosity solution to be at most Lipschitz continuous. Bernstein’s method is usually applied after a change of unknown \(u=\phi (v)\), and the author discusses also the structure conditions needed on \(F\) for this change to be possible. References include 25 items.

35B45 A priori estimates in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35B50 Maximum principles in context of PDEs
35D40 Viscosity solutions to PDEs