Symmetrization on the sphere and applications

Abstract

We introduce a new method of symmetrization of mappings on the $n$-sphere ($n\ge 2$). 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.