Subdifferentials and penalty approximations of the obstacle problem
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916753536188416 |
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| author | Alphonse, Amal Wachsmuth, Gerd |
| author_facet | Alphonse, Amal Wachsmuth, Gerd |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13029 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Subdifferentials and penalty approximations of the obstacle problem Alphonse, Amal Wachsmuth, Gerd Analysis of PDEs Optimization and Control 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. |
| title | Subdifferentials and penalty approximations of the obstacle problem |
| topic | Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2412.13029 |