Bubbling of almost critical points of anisotropic isoperimetric problems with degenerating ellipticity
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914425468878848 |
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| author | Santilli, Mario |
| author_facet | Santilli, Mario |
| contents | Given a sequence of uniformly convex norms $ ϕ_h $ on $ \mathbf{R}^{n+1} $ converging to an arbitrary norm $ ϕ$, we prove rigidity of $ L^1 $-accumulation points of sequences of sets $ E_h \subseteq \mathbf{R}^{n+1} $ of finite perimeter, that are volume-constrained almost-critical points of the anisotropic surface energy functionals associated with $ ϕ_h $. Here, almost criticality is measured in terms of the $ L^n $-deviation from being constant of the distributional anisotropic mean $ ϕ_h $-curvature of (the varifold associated to) of the reduced boundaries of $ E_h $. We prove that such limits are finite union of disjoint, but possibly mutually tangent, $ ϕ$-Wulff shapes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25644 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bubbling of almost critical points of anisotropic isoperimetric problems with degenerating ellipticity Santilli, Mario Analysis of PDEs Differential Geometry Given a sequence of uniformly convex norms $ ϕ_h $ on $ \mathbf{R}^{n+1} $ converging to an arbitrary norm $ ϕ$, we prove rigidity of $ L^1 $-accumulation points of sequences of sets $ E_h \subseteq \mathbf{R}^{n+1} $ of finite perimeter, that are volume-constrained almost-critical points of the anisotropic surface energy functionals associated with $ ϕ_h $. Here, almost criticality is measured in terms of the $ L^n $-deviation from being constant of the distributional anisotropic mean $ ϕ_h $-curvature of (the varifold associated to) of the reduced boundaries of $ E_h $. We prove that such limits are finite union of disjoint, but possibly mutually tangent, $ ϕ$-Wulff shapes. |
| title | Bubbling of almost critical points of anisotropic isoperimetric problems with degenerating ellipticity |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2603.25644 |