Uniqueness of solutions for second order Bellman-Isaacs equations with mixed boundary conditions
In this paper, the authors study the following mixed boundary value problem \[ \begin{aligned} G(x,u,Du,D^2u)=0 &\quad\text{in } {\mathcal O},\\ u=\psi\;\text{ or }\;G(x,u,Du,D^2u)=0 &\quad\text{on } \Gamma_1,\\ G(x,u,Du,D^2u)\leq0 &\quad\text{on } \Gamma_2,\\ u=\psi &\quad\text{on } \Gamma_3, \end{aligned} \] where \(\mathcal O\) is an open set of \({\mathbb R}^n\), which is not necessarily bounded, \(\Gamma_1,\Gamma_2,\Gamma_3\) are pairwise disjoint and \(\Gamma_1\cup\Gamma_2\cup\Gamma_3=\partial{\mathcal O}\), and finally \(G\) is a second order degenerate elliptic operator of Bellman-Isaacs type. The equation and the boundary conditions have to be interpreted in the viscosity sense. Such a problem may arise from exit or stopping time stochastic differential games or optimal control problems with constraints. For such a problem, the authors give a comparison and uniqueness result as well as an existence result. The novelty of those results is due to the presence of the three different boundary conditions in connection with the fact that \(G\) is a second order operator.
scientific article; zbMATH DE number 5347944