Semiregular and strongly irregular boundary points for nonlocal Dirichlet problems
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915364075470848 |
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| author | Björn, Anders Björn, Jana Kim, Minhyun |
| author_facet | Björn, Anders Björn, Jana Kim, Minhyun |
| contents | In this paper we study nonlocal nonlinear equations of fractional $(s,p)$-Laplacian type on $\mathbf{R}^n$. We show that the irregular boundary points for the Dirichlet problem can be divided into two disjoint classes: semiregular and strongly irregular boundary points, with very different behaviour. Two fundamental tools needed to show this are the Kellogg property (from our previous paper) and a new removability result for solutions in the $V^{s,p}$ Sobolev type space, which we deduce more generally also for supersolutions of equations with a right-hand side. Semiregular and strongly irregular points are also characterized in various ways. Finally, it is explained how semiregularity depends on $s$ and $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23188 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Semiregular and strongly irregular boundary points for nonlocal Dirichlet problems Björn, Anders Björn, Jana Kim, Minhyun Analysis of PDEs Primary: 35R11. Secondary: 31C15, 31C45, 35J66 In this paper we study nonlocal nonlinear equations of fractional $(s,p)$-Laplacian type on $\mathbf{R}^n$. We show that the irregular boundary points for the Dirichlet problem can be divided into two disjoint classes: semiregular and strongly irregular boundary points, with very different behaviour. Two fundamental tools needed to show this are the Kellogg property (from our previous paper) and a new removability result for solutions in the $V^{s,p}$ Sobolev type space, which we deduce more generally also for supersolutions of equations with a right-hand side. Semiregular and strongly irregular points are also characterized in various ways. Finally, it is explained how semiregularity depends on $s$ and $p$. |
| title | Semiregular and strongly irregular boundary points for nonlocal Dirichlet problems |
| topic | Analysis of PDEs Primary: 35R11. Secondary: 31C15, 31C45, 35J66 |
| url | https://arxiv.org/abs/2506.23188 |