Optimal regularity in the optimal switching problem

TL;DR

This paper establishes the optimal regularity (C^{1,1}) for solutions to a coupled obstacle problem in the context of optimal switching, under specific assumptions about the zero loop set, and demonstrates the necessity of these assumptions.

Contribution

It proves the optimal C^{1,1} regularity for solutions to a weakly coupled obstacle system in optimal switching, under a key geometric assumption on the zero loop set.

Findings

Proves C^{1,1} regularity under the zero loop set assumption.

Shows the regularity result is optimal with a counterexample.

Highlights the importance of the zero loop set's interior closure in regularity.

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 (-\Delta u^1+f^1, u^1-u^2+\psi^1)=0 \\ \min (-\Delta u^2+f^2, u^2-u^1+\psi^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 $\mathscr{L}:= \{ \psi^1+\psi^2=0\}$ 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 $\mathscr{L} = \overline{ \mathscr{L}^0} $.