A theorem on degenerate elliptic Bellman equations in bounded domains

Abstract

A general existence and uniqueness theorem for degenerate elliptic Bellman equations in bounded domains is proved. Functional classes $C^{2+\alpha}(D)$ and $C^{1,1}(D)$ are the classes where solutions are looked for. This theorem has a very broad range of applicability. Equations $u_{t}=u_{xx}$ and $$ P_{m}(u_{xx})=\sum_{k=0}^{m-1}c_{k}(x)P_{k}(u_{xx}), $$ where $P_{k}(u_{xx})$ is the $k$th elementary symmetric polynomial of eigenvalues of the matrix $u_{xx}$ are particular cases of equations under consideration.