Superposition in the $p$-Laplace Equation

TL;DR

This paper provides a simple proof that superpositions of fundamental solutions to the p-Laplace equation are p-superharmonic, extending the known results with an explicit formula for the p-Laplacian of such superpositions.

Contribution

It offers a straightforward proof and extends the understanding of superpositions in the p-Laplace equation using an explicit formula.

Findings

Superpositions of fundamental solutions are p-superharmonic for p>2.

Provided an explicit formula for the p-Laplacian of superpositions.

Extended previous results with a simpler proof.

Abstract

That a superposition of fundamental solutions to the $p$-Laplace Equation is $p$-superharmonic -- even in the non-linear cases $p>2$ -- has been known since M. Crandall and J. Zhang published their paper "Another Way to Say Harmonic" in 2003. We give a simple proof and extend the result by means of an explicit formula for the $p$-Laplacian of the superposition.