Gradient H\"{o}lder regularity for nonlocal double phase equations

TL;DR

This paper establishes interior $C^{1, eta}$ regularity for viscosity solutions to nonlocal double phase equations, highlighting the influence of coefficient regularity and growth exponents.

Contribution

It proves gradient Hölder continuity for solutions under Lipschitz conditions on the coefficient, addressing a higher regularity issue in degenerate cases.

Findings

Gradient of solutions is Hölder continuous under certain conditions.

Regularity depends on the Lipschitz continuity of the coefficient.

The work addresses a higher regularity problem raised in prior research.

Abstract

This paper is devoted to investigating the interior $C^{1, \alpha}$ 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 H\"older 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.