Interior $C^{1,α}$ regularity of mixed local-nonlocal $(p,q)$-energy minimizers for $p\leq sq$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917132897353728 |
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| author | Biswas, Anup Topp, Erwin |
| author_facet | Biswas, Anup Topp, Erwin |
| contents | We establish the local $C^{1, α}$ regularity of minimizers for functionals of the form $$w\to \int_Ω(|\nabla w|^p-fw) dx + \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{|w(x)-w(y)|^q}{|x-y|^{n+sq}}dx\, dy,$$ where $s \in (0, 1)$, $1 < p \leq sq$, and $f \in L^\infty(Ω)$. This result complements the work of De Filippis and Minigione in \cite{DFM}, thereby completing the proof of $C^{1,α}$ regularity for all $p, q \in (1, \infty)$ and $s \in (0, 1)$ with locally bounded source term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_07481 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Interior $C^{1,α}$ regularity of mixed local-nonlocal $(p,q)$-energy minimizers for $p\leq sq$ Biswas, Anup Topp, Erwin Analysis of PDEs We establish the local $C^{1, α}$ regularity of minimizers for functionals of the form $$w\to \int_Ω(|\nabla w|^p-fw) dx + \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{|w(x)-w(y)|^q}{|x-y|^{n+sq}}dx\, dy,$$ where $s \in (0, 1)$, $1 < p \leq sq$, and $f \in L^\infty(Ω)$. This result complements the work of De Filippis and Minigione in \cite{DFM}, thereby completing the proof of $C^{1,α}$ regularity for all $p, q \in (1, \infty)$ and $s \in (0, 1)$ with locally bounded source term. |
| title | Interior $C^{1,α}$ regularity of mixed local-nonlocal $(p,q)$-energy minimizers for $p\leq sq$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.07481 |