Local H\"older regularity for bounded, signed solutions to nonlocal Trudinger equations

Abstract

We prove local H\"older regularity for bounded and sign-changing weak solutions to nonlocal Trudinger equations of the form \[ (|u|^{p-2}u)_t + \text{P.V.} \int_{\mathbb{R}^n} \frac{|u(x,t) - u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}} = 0, \] in the range $1< p<\infty$ and $s \in (0,1)$. One of the main difficulties in extending the local theory to the nonlocal Trudinger equation is that when $0 \ll u \ll \infty$ locally, a crucial change of variable is unavailable in the nonlocal case due to the presence of the Tail term. We adapt several new ideas developed in the past few years to prove the required H\"older regularity.