A priori Lipschitz estimates for nonlinear equations with mixed local and nonlocal diffusion via the adjoint-Bernstein method

TL;DR

This paper develops a new method to derive Lipschitz estimates for nonlinear equations involving both local and nonlocal diffusion, using an advanced Bernstein approach and a nonlocal Bochner identity.

Contribution

It introduces an integral refinement of the Bernstein method combined with a nonlocal Bochner identity for the first time in this context.

Findings

Established a priori Lipschitz estimates for mixed diffusion equations

Extended Bernstein method to nonlocal and nonlinear settings

Provided tools for analyzing equations with unbounded right-hand sides

Abstract

We establish a priori Lipschitz estimates for equations with mixed local and nonlocal diffusion, coercive gradient terms and unbounded right-hand side in Lebesgue spaces through an integral refinement of the Bernstein method. This relies on a nonlinear, nonlocal and variational version of the Bochner identity that involves the adjoint equation of the linearization of the initial problem.