Perimeter on a manifold, with applications to partial differential equations

Satyanad Kichenassamy

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arXiv · 2507.04740

Perimeter on a manifold, with applications to partial differential equations

Satyanad Kichenassamy

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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.

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Satyanad Kichenassamy. 2025-07-07. Perimeter on a manifold, with applications to partial differential equations. https://arxiv.org/abs/2507.04740

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