$C^\infty$ regularity in semilinear free boundary problems
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916339182993408 |
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| author | Restrepo, Daniel Ros-Oton, Xavier |
| author_facet | Restrepo, Daniel Ros-Oton, Xavier |
| contents | We study the higher regularity of solutions and free boundaries in the Alt-Phillips problem $Δu=u^{γ-1}$, with $γ\in(0,1)$. Our main results imply that, once free boundaries are $C^{1,α}$, then they are $C^\infty$. In addition $u/d^{\frac{2}{2-γ}}$ and $u^{\frac{2-γ}{2}}$ are $C^\infty$ too.
In order to achieve this, we need to establish fine regularity estimates for solutions of linear equations with boundary-singular Hardy potentials $-Δv = κv/d^2$ in $Ω$, where $d$ is the distance to the boundary and $κ\leq\frac{1}{4}$. Interestingly, we need to include even the critical constant $κ=\frac{1}{4}$, which corresponds to $γ=\frac{2}{3}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20426 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $C^\infty$ regularity in semilinear free boundary problems Restrepo, Daniel Ros-Oton, Xavier Analysis of PDEs We study the higher regularity of solutions and free boundaries in the Alt-Phillips problem $Δu=u^{γ-1}$, with $γ\in(0,1)$. Our main results imply that, once free boundaries are $C^{1,α}$, then they are $C^\infty$. In addition $u/d^{\frac{2}{2-γ}}$ and $u^{\frac{2-γ}{2}}$ are $C^\infty$ too. In order to achieve this, we need to establish fine regularity estimates for solutions of linear equations with boundary-singular Hardy potentials $-Δv = κv/d^2$ in $Ω$, where $d$ is the distance to the boundary and $κ\leq\frac{1}{4}$. Interestingly, we need to include even the critical constant $κ=\frac{1}{4}$, which corresponds to $γ=\frac{2}{3}$. |
| title | $C^\infty$ regularity in semilinear free boundary problems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.20426 |