Subdifferentials and penalty approximations of the obstacle problem

TL;DR

This paper develops a theoretical framework for approximating the obstacle problem using nonlinear PDEs with penalty methods, analyzing subdifferential convergence and implications for optimal control.

Contribution

It introduces a unified approach to analyze penalty approximations of the obstacle problem via capacity theory and subdifferential convergence, applicable to smooth and nonsmooth penalties.

Findings

Derivatives of penalized solutions converge in the weak operator topology.

The framework applies to common penalty functions including nonsmooth ones.

Implications for the optimal control theory of the obstacle problem.

Abstract

We consider a framework for approximating the obstacle problem through a penalty approach by nonlinear PDEs. By using tools from capacity theory, we show that derivatives of the solution maps of the penalised problems converge in the weak operator topology to an element of the strong-weak Bouligand subdifferential. We are able to treat smooth penalty terms as well as nonsmooth ones involving for example the positive part function $\max(0,\cdot)$. Our abstract framework applies to several specific choices of penalty functions which are omnipresent in the literature. We conclude with consequences to the theory of optimal control of the obstacle problem.