Gradient Hölder regularity in mixed local and nonlocal linear parabolic problem

Abstract

We prove the local Hölder regularity of weak solutions to the mixed local nonlocal parabolic equation of the formut−Δu+P.V.∫Rnu(x,t)−u(y,t)|x−y|n+2sdy=0, where 0<s<1; for every exponent α0∈(0,1). Here, Δ is the usual Laplace operator. Next, we show that the gradients of weak solutions are also α-Hölder continuous for some α∈(0,1). Our approach is purely analytic and it is based on perturbation techniques.