A variational inequality approach to the Bellman-Dirichlet equation for two elliptic operators

Abstract

We prove existence, uniqueness, and regularity properties for a solution u of the Bellman-Dirichlet equation of dynamic programming:

$$\left\{ \begin{gathered} \max {\text{ }}\{ L^i u + f^i = 0{\text{ in }}\Omega \hfill \\ i{\text{ = 1,2 }} \hfill \\ u{\text{ = 0 on }}\partial \Omega , \hfill \\ \end{gathered} \right.$$

((1))

where L ¹ and L ² are two second order, uniformly elliptic operators. The method of proof is to rephrase (1) as a variational inequality for the operator K=L ²(L ¹)⁻¹ in L ²(Ω) and to invoke known existence theorems. For sufficiently nice f ¹ and f ² we prove in addition that u is in H ³(Ω)⋂C ^2,α(Ω) (for some 0<α<1) and hence is a classical solution of (1).