Gradient regularity for nonlocal double phase equations

2026/04/24 by Yuzhou Fang, Chao Zhang
#math.AP

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Abstract

This paper is devoted to investigating the interior C^{1, α} regularity of viscosity solutions to the nonlocal double phase equations ∫_ℝ^d (\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)dy=0, where 2≤ p≤ q, 0<s≤ t<1, and a(x, y)≥0. By assuming the Lipschitz continuity of a(⋅), 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(⋅), 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(⋅), the growth exponents and the differentiability orders.

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