Subdifferentials and penalty approximations of the obstacle problem

Autor:	Alphonse, Amal, Wachsmuth, Gerd
Rok vydání:	2024
Předmět:	Mathematics - Analysis of PDEs Mathematics - Optimization and Control
Druh dokumentu:	Working Paper
Popis:	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.
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2412.13029 Zobrazit plný text záznamu View this record from Arxiv