Lipschitz regularity for fractional $p$-Laplacian with coercive gradients

TL;DR

This paper proves local Lipschitz regularity and H"older continuity for solutions of certain nonlinear nonlocal equations involving fractional p-Laplacian operators with coercive gradient terms.

Contribution

It establishes Lipschitz regularity for solutions under specific parameter ranges and shows triviality of bounded solutions when the forcing term is zero and the Hamiltonian is spatially independent.

Findings

Viscosity solutions are locally Lipschitz continuous under given conditions.

Subsolutions are H"older continuous.

Only trivial bounded solutions exist when f=0 and H is independent of x.

Abstract

In this article, we study nonlinear nonlocal equations with coercive gradient nonlinearity of the form \[ (-\Delta_p)^s u(x) + H(x, \nabla u) = f, \] where $f$ is Lipschitz continuous. We show that any viscosity solution $u$ is locally Lipschitz continuous, provided \[ p \in \left(1, \frac{2}{1-s}\right) \cup (1, m+1). \] We also establish H\"older continuity of subsolutions. Furthermore, in the case $f=0$ and $H$ is independent of $x$, we prove that the equation admits only the trivial solution in the class of bounded solutions, for all $m, p \in (1,\infty)$.