Gradient regularity for nonlocal double phase equations

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 $2\le p\le q$, $0<s\le t<1$, and $a(x,y)\ge 0$. By assuming the Lipschitz continuity of $a(\cdot )$, we show that the gradient of solution is Hölder continuous, provided the distance of $tq$ and $sp$ is suitably small. As a key ingredient to this conclusion, the Lipschitz property of solutions is also established under weaker assumptions on the modulating coefficient $a(\cdot )$, which is of independent interest. Our results develop a nonlocal counterpart of the gradient regularity theory for classical double phase problems due to Colombo \& Mingione [Arch. Ration. Mech. Anal., 2015] and solve the higher regularity issue raised by De Filippis \& Palatucci [J. Differential Equations, 2019]. The core challenges consist in precisely characterizing the subtle interaction among the pointwise behaviour of the coefficient $a(\cdot )$, the growth exponents and the differentiability orders.