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Convex viscosity solutions and state constraints. (English) Zbl 0890.49013
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.

MSC:
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35J65 Nonlinear boundary value problems for linear elliptic equations
35J60 Nonlinear elliptic equations
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