Quantitative results for fractional overdetermined problems in exterior and annular sets
Abstract
We consider overdetermined problems related to the fractional capacity. In particular we study $s$-harmonic functions defined in unbounded exterior sets or in bounded annular sets, and having a level set parallel to the boundary. We first classify the solutions of the overdetermined problems, by proving that the domain and the solution itself are radially symmetric. Then we prove a quantitative stability counterpart of the symmetry results: we assume that the overdetermined condition is slightly perturbed and we measure, in a quantitative way, how much the domain is close to a symmetric set.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2022
- DOI:
- 10.48550/arXiv.2209.14679
- arXiv:
- arXiv:2209.14679
- Bibcode:
- 2022arXiv220914679C
- Keywords:
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- Mathematics - Analysis of PDEs