Perimeter on a manifold, with applications to partial differential equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915373556695040 |
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| author | Kichenassamy, Satyanad |
| author_facet | Kichenassamy, Satyanad |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_04740 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Perimeter on a manifold, with applications to partial differential equations Kichenassamy, Satyanad Analysis of PDEs Differential Geometry 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. |
| title | Perimeter on a manifold, with applications to partial differential equations |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2507.04740 |