Perimeter on a manifold, with applications to partial differential equations

TL;DR

This paper extends the concept of perimeter to subsets of Riemannian manifolds, establishes its properties via heat kernel regularization, and applies these results to solve a quasilinear elliptic PDE using a novel symmetrization method.

Contribution

It generalizes perimeter to Riemannian manifolds, proves its relation to heat kernel regularization, and introduces a new symmetrization technique for PDEs where traditional methods fail.

Findings

Perimeter defined as heat kernel regularization limit on manifolds

Generalized isoperimetric inequality and Fleming-Rishel formula

Successfully applied symmetrization on the sphere to a PDE problem

Abstract

The perimeter of a measurable subset of $\mathbb R^N$ is the total variation of its characteristic function. We generalize this notion to a subset $E$ of a closed Riemannian manifold. We show that the perimeter of $E$ is the limit of the hear kernel regularization of its characteristic function. A generalization of the isoperimetric inequality and of the Fleming-Rishel formula follow. These results are applied to a quasilinear elliptic problem in $\mathbb R^N$ for which the usual symmetrization methods fail. It will be tackled successfully by introducing a symmetrization method on the sphere.