On a particular scaling for the prototype anisotropic p-Laplacian

TL;DR

This paper demonstrates that a specific volume non-preserving scaling enables the development of a regularity theory for local weak solutions to a fully anisotropic parabolic p-Laplacian equation, highlighting self-similarity and invariance properties.

Contribution

It introduces a novel scaling approach that facilitates regularity analysis and characterizes self-similar solutions for the anisotropic p-Laplacian.

Findings

Scaling preserves semi-continuity of solutions

Self-similar solutions are characterized under the new scaling

Regularity theory can be developed using this scaling approach

Abstract

In this brief note we show that under a volume non-preserving scaling it is possible to recover the basics for a regularity theory regarding local weak solutions to a parabolic fully anisotropic equation. We characterize self-similar solutions regarding this particular scaling and we show that semi-continuity for solutions to this equation is a consequence of a simple property that is itself invariant under scaling.