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 \begin{equation*} \begin{cases} \min (-Δu^1+f^1, u^1-u^2+ψ^1)=0 \\ \min (-Δu^2+f^2, u^2-u^1+ψ^2)=0, \end{cases} \end{equation*} 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 $\mathcal{L}=\overline{{\mathcal{L}}^0}$.

19 pages, preprint

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