Optimal regularity in the optimal switching problem

Abstract

In this article we study the optimal regularity for solutions to the following weakly coupled system with interconnected obstacles

arising in the optimal switching problem with two modes.

We derive the optimal $C^{1,1}$-regularity for the minimal solution under the assumption that the zero loop set $\mathcal{L}:=\{{\psi }^1+{\psi }^2=0\}$ is the closure of its interior. This result is optimal and we provide a counterexample showing that the $C^{1,1}$-regularity does not hold without the assumption $\mathcal{L}=\overline{{\mathcal{L}}^0}$.