Convex viscosity solutions and state constraints (Q1357069)

The authors study the viscosity solution of some general second-order fully nonlinear elliptic equation with state constraint boundary conditions. They prove that a viscosity solution is convex, by combining a comparison principle with the fact that the convex envelope of the solution is a supersolution. The last property relies on a characterization of the viscosity subject of the convex envelope of a lower semicontinuous coercive function. The equation solved by the conjugate of a convex solution is studied and some questions of partial convexity are considered.