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ON THE DIRICHLET PROBLEM FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS AND APPLICATIONS TO OPTIMAL CONTROL

We construct a generalized viscosity solution of the Dirichlet problem for fully nonlinear degenerate elliptic equations in general domains by the Perron-WienerBrelot method. The result is designed for the Hamilton-Jacobi-Bellman-Isaacs equations of time-optimal stochastic control and differential games with discontinuous value function. We study several properties of the generalized solution, in particular its approximation via vanishing viscosity and r egularization of the domain. The connection with optimal control is proved for a deterministic minimumtime problem and for the problem of maximizing the expected escape time of a degenerate diffusion process from an open set.