Perimeter on a manifold, with applications to partial differential equations

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.

in French language, S{é}minaire E.D.P., dit aussi ''S{é}minaire Goulaouic-Schwartz'', 1986-1987, Ecole Polytechnique., 1987, Palaiseau, France