Symmetrization on the sphere and applications

TL;DR

This paper introduces a new symmetrization method on the sphere and applies it to estimate solutions of certain quasilinear elliptic PDEs, especially those involving the p-Laplacian, with specific focus on the case p=n.

Contribution

The paper presents a novel symmetrization technique on the sphere and demonstrates its application to estimate solutions of p-Laplacian type equations with measure data.

Findings

Effective symmetrization method for sphere mappings.

Application to p-Laplacian PDEs with measure data.

Reduction of the p=n case to a conformal problem on the sphere.

Abstract

We introduce a new method of symmetrization of mappings on the $n$-sphere ($n\geq 2$). They are applied to estimate solutions of quasilinear elliptic partial differential equations of $p$-Laplacian type, with combinations of Dirac measures on the right-hand side. The case $p=n$ is reduced to a problem on the sphere, using a conformal transformation. The cases when $1 < p < n $ and $p > $n are considered more briefly, full details being available in other papers of the author.