Optimal stochastic switching and the Dirichlet problem for the Bellman equation

Let L i {L^i} be a sequence of second order elliptic operators in a bounded n -dimensional domain Ω Ω , and let f i {f^i} be given functions. Consider the problem of finding a solution u to the Bellman equation sup i ( L i u − f i ) = 0 { _i}({L^i}u\, - \,{f^i})\, = \,0 a.e. in Ω Ω , subject to the Dirichlet boundary condition u = 0 u\, = \,0 on ∂ Ω ∂ Ω . It is proved that, provided the leading coefficients of the L i {L^i} are constants, there exists a unique solution u of this problem, belonging to W 1 , ∞ ( Ω ) ∩ W loc 2 ,