Concavity estimates for a class of nonlinear elliptic equations in two dimensions

We study solutions of the nonlinear elliptic equation upsilonDeltaupsilon = C-2 + C-1\Dupsilon\(2) on a bounded domain Omega in R-2. It is shown that the set M_ of points where the graph of the solution has negative Gauss curvature always extends to the boundary, unless it is empty. The meethod uses an elliptic equation satisfied by an auxiliary function given by the product of the Hessian determinant and a suitable power of the solutions. As a consequence of the result, we give a new proof for power concavity of solutions to certain semilinear boundary value problems in convex domains.