Quantitative estimates for the relative isoperimetric problem and its gradient flow outside convex bodies in the plane
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912739178315776 |
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| author | Mäder-Baumdicker, Elena Neumayer, Robin Park, Jiewon Rupflin, Melanie |
| author_facet | Mäder-Baumdicker, Elena Neumayer, Robin Park, Jiewon Rupflin, Melanie |
| contents | We prove three related quantitative results for the relative isoperimetric problem outside a convex body $Ω$ in the plane: (1) Łojasiewicz estimates and quantitative rigidity for critical points, (2) rates of convergence for the gradient flow, and (3) quantitative stability for minimizers. These results come with explicit constants and optimal exponents/rates, and hold whenever a simple two-dimensional auxiliary variational problem for circular arcs outside of $Ω$ is nondegenerate. The proofs are inter-related, and in particular, for the first time in the context of isoperimetric problems, a flow approach is used to prove quantitative stability for minimizers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_21198 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative estimates for the relative isoperimetric problem and its gradient flow outside convex bodies in the plane Mäder-Baumdicker, Elena Neumayer, Robin Park, Jiewon Rupflin, Melanie Analysis of PDEs Differential Geometry 58E35, 53E10, 49Q10 We prove three related quantitative results for the relative isoperimetric problem outside a convex body $Ω$ in the plane: (1) Łojasiewicz estimates and quantitative rigidity for critical points, (2) rates of convergence for the gradient flow, and (3) quantitative stability for minimizers. These results come with explicit constants and optimal exponents/rates, and hold whenever a simple two-dimensional auxiliary variational problem for circular arcs outside of $Ω$ is nondegenerate. The proofs are inter-related, and in particular, for the first time in the context of isoperimetric problems, a flow approach is used to prove quantitative stability for minimizers. |
| title | Quantitative estimates for the relative isoperimetric problem and its gradient flow outside convex bodies in the plane |
| topic | Analysis of PDEs Differential Geometry 58E35, 53E10, 49Q10 |
| url | https://arxiv.org/abs/2508.21198 |