Parabolic theory of the discrete p-Laplace operator

TL;DR

This paper investigates the discrete p-Laplacian operator, exploring its variational properties and implications for parabolic equations on graphs, including positivity, comparison principles, and symmetry considerations.

Contribution

It introduces a variational framework for the discrete p-Laplacian and analyzes its impact on parabolic problems, highlighting new insights into nonlinear Laplacians on graphs.

Findings

Established positivity and comparison principles for the discrete p-Laplacian

Analyzed symmetry compatibility in graph-based operators

Discussed variational properties of nonlinear generalized Laplacians

Abstract

We study the discrete version of the $p$-Laplacian. Based on its variational properties we discuss some features of the associated parabolic problem. Our approach allows us in turn to obtain interesting information about positivity and comparison principles as well as compatibility with the symmetries of the graph. We conclude briefly discussing the variational properties of a handful of nonlinear generalized Laplacians appearing in different parabolic equations.