A Dirichlet Type Problem for Nonlinear Degenerate Elliptic Equations Arising in Time-optimal Stochastic Control
نویسنده
چکیده
We study general 2nd order fully nonlinear degenerate elliptic equations on an arbitrary closed set with generalized Dirichlet boundary conditions in the viscosity sense. We prove some properties of the maximal subsolution and the minimal supersolution of the Dirichlet type problem. Under a sort of compatibility condition on the boundary data we show that the maximal subsolution is the natural generalized solution of the boundary value problem, even if it is not necessarily continuous, and we give approximation theorems, in particular by penalization. We apply these results to the Hamilton-Jacobi-Bellman-Isaacs equations of time-optimal stochastic control and pursuit-evasion games whose value functions, which are discontinuous in general, are characterized as the unique solution in the previous sense of the appropriate Dirichlet type problem. 1 Partially supported by M.U.R.S.T. project "Problemi nonlineari nell'analisi e nelle appli-cazioni siche, chimiche, biologiche".
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