Gradient regularity for $(s,p)$-harmonic functions
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914934910091264 |
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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 | We study the local regularity properties of $(s,p)$-harmonic functions, i.e. local weak solutions to the fractional $p$-Laplace equation of order $s\in (0,1)$ in the case $p\in (1,2]$. It is shown that $(s,p)$-harmonic functions are weakly differentiable and that the weak gradient is locally integrable to any power $q\geq 1$. As a result, $(s,p)$-harmonic functions are Hölder continuous to arbitrary Hölder exponent in $(0,1)$. In addition, the weak gradient of $(s,p)$-harmonic functions has certain fractional differentiability. All estimates are stable when $s$ reaches $1$, and the known regularity properties of $p$-harmonic functions are formally recovered, in particular the local $W^{2,2}$-estimate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02012 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gradient regularity for $(s,p)$-harmonic functions Bögelein, Verena Duzaar, Frank Liao, Naian Bisci, Giovanni Molica Servadei, Raffaella Analysis of PDEs We study the local regularity properties of $(s,p)$-harmonic functions, i.e. local weak solutions to the fractional $p$-Laplace equation of order $s\in (0,1)$ in the case $p\in (1,2]$. It is shown that $(s,p)$-harmonic functions are weakly differentiable and that the weak gradient is locally integrable to any power $q\geq 1$. As a result, $(s,p)$-harmonic functions are Hölder continuous to arbitrary Hölder exponent in $(0,1)$. In addition, the weak gradient of $(s,p)$-harmonic functions has certain fractional differentiability. All estimates are stable when $s$ reaches $1$, and the known regularity properties of $p$-harmonic functions are formally recovered, in particular the local $W^{2,2}$-estimate. |
| title | Gradient regularity for $(s,p)$-harmonic functions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.02012 |