Gradient Hölder regularity for nonlocal double phase equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914504471740416 |
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| author | Fang, Yuzhou Zhang, Chao |
| author_facet | Fang, Yuzhou Zhang, Chao |
| contents | This paper is devoted to investigating the interior $C^{1, α}$ regularity of viscosity solutions to the nonlocal double phase equations $$ \int_{\mathbb{R}^d} \left(\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+sp}}+a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{d+tq}}\right)\,dy=0, $$ where $2\le p\le q$, $s, t\in (0, 1)$ with $s\le t$, and $a(x, y)\ge0$. In the degenerate case, we solve the higher regularity issue raised by De Filippis-Palatucci [J. Differential Equations \textbf{267} (2019) 547--586]. By assuming the Lipschitz continuity of the modulating coefficient $a$, we are able to prove that the gradient of solution is Hölder continuous, provided the distance of $tq$ and $sp$ is suitably small. The core challenges consist in precisely characterizing the subtle interaction among the pointwise behaviour of the coefficient $a$, the growth exponents and the differentiability orders. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22206 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gradient Hölder regularity for nonlocal double phase equations Fang, Yuzhou Zhang, Chao Analysis of PDEs This paper is devoted to investigating the interior $C^{1, α}$ regularity of viscosity solutions to the nonlocal double phase equations $$ \int_{\mathbb{R}^d} \left(\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+sp}}+a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{d+tq}}\right)\,dy=0, $$ where $2\le p\le q$, $s, t\in (0, 1)$ with $s\le t$, and $a(x, y)\ge0$. In the degenerate case, we solve the higher regularity issue raised by De Filippis-Palatucci [J. Differential Equations \textbf{267} (2019) 547--586]. By assuming the Lipschitz continuity of the modulating coefficient $a$, we are able to prove that the gradient of solution is Hölder continuous, provided the distance of $tq$ and $sp$ is suitably small. The core challenges consist in precisely characterizing the subtle interaction among the pointwise behaviour of the coefficient $a$, the growth exponents and the differentiability orders. |
| title | Gradient Hölder regularity for nonlocal double phase equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.22206 |