Generalized derivatives for the solution operator of the obstacle problem

TL;DR

This paper characterizes generalized derivatives of the obstacle problem's solution operator using capacitary measures, enabling a new optimality condition for control problems that surpasses existing C-stationarity conditions.

Contribution

It introduces a novel characterization of generalized derivatives for the obstacle problem's solution operator using capacitary measures, leading to stronger optimality conditions.

Findings

New characterization of generalized derivatives using capacitary measures

Development of a stronger optimality condition for obstacle control problems

Comparison showing improvement over C-stationarity conditions

Abstract

We characterize generalized derivatives of the solution operator of the obstacle problem. This precise characterization requires the usage of the theory of so-called capacitary measures and the associated solution operators of relaxed Dirichlet problems. The generalized derivatives can be used to obtain a novel necessary optimality condition for the optimal control of the obstacle problem with control constraints. A comparison shows that this system is stronger than the known system of C-stationarity.