Comparison Principles for Second Order Elliptic/Parabolic Equations with Discontinuities in the Gradient Compatible with Finsler Norms

TL;DR

This paper develops new comparison principles for second order elliptic and parabolic PDEs with gradient discontinuities aligned with Finsler norms, unifying and extending previous results in the field.

Contribution

It introduces a unified framework for comparison principles applicable to PDEs with Finsler-compatible gradient discontinuities, generalizing prior approaches.

Findings

New comparison results for PDEs with Finsler-compatible gradient discontinuities

Unified framework encompassing previous special cases

Applications to surface growth models and variational problems

Abstract

This paper is about elliptic and parabolic partial differential operators with discontinuities in the gradient which are compatible with a Finsler norm in a sense to be made precise. Examples of this type of problems arise in a number of contexts, most notably the recent work of Chatterjee and the second author [7] on scaling limits of discrete surface growth models as well as $L^{\infty}-$variational problems. Building on the approach of Ishii [16], new comparison results are proven within a unified framework that includes a number of previous results as special cases.