Free energy minimizers with radial densities: classification and quantitative stability
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915127693934592 |
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| author | Aryan, Shrey Silini, Lauro |
| author_facet | Aryan, Shrey Silini, Lauro |
| contents | We study the isoperimetric problem with a potential energy $g$ in $\mathbb{R}^n$ weighted by a radial density $f$ and analyze the geometric properties of minimizers. Notably, we construct two counterexamples demonstrating that, in contrast to the classical isoperimetric case $g = 0$, the condition $\ln(f)'' + g' \geq 0$ does not generally guarantee the global optimality of centered spheres. However, we demonstrate that centered spheres are globally optimal when both $f$ and $g$ are monotone. Additionally, we strengthen this result by deriving a sharp quantitative stability inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03997 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Free energy minimizers with radial densities: classification and quantitative stability Aryan, Shrey Silini, Lauro Analysis of PDEs We study the isoperimetric problem with a potential energy $g$ in $\mathbb{R}^n$ weighted by a radial density $f$ and analyze the geometric properties of minimizers. Notably, we construct two counterexamples demonstrating that, in contrast to the classical isoperimetric case $g = 0$, the condition $\ln(f)'' + g' \geq 0$ does not generally guarantee the global optimality of centered spheres. However, we demonstrate that centered spheres are globally optimal when both $f$ and $g$ are monotone. Additionally, we strengthen this result by deriving a sharp quantitative stability inequality. |
| title | Free energy minimizers with radial densities: classification and quantitative stability |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.03997 |