WIAS Preprint No. 3159, (2025)

Subdifferentials and penalty approximations of the obstacle problem


Authors

  • Alphonse, Amal
    ORCID: 0000-0001-7616-3293
  • Wachsmuth, Gerd

2020 Mathematics Subject Classification

  • 49J40 65K15 49K21 49M41 47B92 28A12

Keywords

  • Variational inequalities, obstacle problem, optimal control, weak operator topology, measures

DOI

10.20347/WIAS.PREPRINT.3159

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, ·). 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.

Download Documents