Regularity for the fractional $p$-Laplace equation
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909215572885504 |
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| author | Bögelein, Verena Duzaar, Frank Liao, Naian Bisci, Giovanni Molica Servadei, Raffaella |
| author_facet | Bögelein, Verena Duzaar, Frank Liao, Naian Bisci, Giovanni Molica Servadei, Raffaella |
| contents | Higher Sobolev and Hölder regularity is studied for local weak solutions of the fractional $p$-Laplace equation of order $s$ in the case $p\ge 2$. Depending on the regime considered, i.e. $$0<s\le\tfrac{p-2}{p}\quad \text{or} \quad\tfrac{p-2}{p}<s<1,$$ precise local estimates are proven. The relevant estimates are stable if the fractional order $s$ reaches $1$; the known Sobolev regularity estimates for the local $p$-Laplace are recovered. The case $p=2$ reproduces the almost $W^{1+s,2}_{\rm loc}$-regularity for the fractional Laplace equation of any order $s\in(0,1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01568 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularity for the fractional $p$-Laplace equation Bögelein, Verena Duzaar, Frank Liao, Naian Bisci, Giovanni Molica Servadei, Raffaella Analysis of PDEs Higher Sobolev and Hölder regularity is studied for local weak solutions of the fractional $p$-Laplace equation of order $s$ in the case $p\ge 2$. Depending on the regime considered, i.e. $$0<s\le\tfrac{p-2}{p}\quad \text{or} \quad\tfrac{p-2}{p}<s<1,$$ precise local estimates are proven. The relevant estimates are stable if the fractional order $s$ reaches $1$; the known Sobolev regularity estimates for the local $p$-Laplace are recovered. The case $p=2$ reproduces the almost $W^{1+s,2}_{\rm loc}$-regularity for the fractional Laplace equation of any order $s\in(0,1)$. |
| title | Regularity for the fractional $p$-Laplace equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.01568 |