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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2310.13656 |
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| _version_ | 1866915254121791488 |
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| author | Bucur, Claudia |
| author_facet | Bucur, Claudia |
| contents | In this paper, we study the existence of solutions of the equation $(-Δ)_1^s u=f$ in a bounded open set with Lipschitz boundary $Ω\subset \Rn$, vanishing on $\Co Ω$, for some given $s\in (0,1)$, and asymptotics as $p\to 1$ of solutions of $(-Δ)_p^s u=f$. We obtain existence and convergence by comparing the $L^{\frac{n}{s}}$ norm of $f$ to the sharp fractional Sobolev constant, or, when $f$ is non-negative, the weighted fractional Cheegar constant to $1$ -- in this case, the results are sharp. We further prove that solutions are "flat" on sets of positive Lebesgue measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_13656 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Solutions of the fractional 1-Laplacian: existence, asymptotics and flatness results Bucur, Claudia Analysis of PDEs In this paper, we study the existence of solutions of the equation $(-Δ)_1^s u=f$ in a bounded open set with Lipschitz boundary $Ω\subset \Rn$, vanishing on $\Co Ω$, for some given $s\in (0,1)$, and asymptotics as $p\to 1$ of solutions of $(-Δ)_p^s u=f$. We obtain existence and convergence by comparing the $L^{\frac{n}{s}}$ norm of $f$ to the sharp fractional Sobolev constant, or, when $f$ is non-negative, the weighted fractional Cheegar constant to $1$ -- in this case, the results are sharp. We further prove that solutions are "flat" on sets of positive Lebesgue measure. |
| title | Solutions of the fractional 1-Laplacian: existence, asymptotics and flatness results |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2310.13656 |