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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2402.04675 |
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| _version_ | 1866912050814386176 |
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| author | Pascale, Giulio Pozzetta, Marco |
| author_facet | Pascale, Giulio Pozzetta, Marco |
| contents | We consider capillarity functionals which measure the perimeter of sets contained in a Euclidean half-space assigning a constant weight $λ\in (-1,1)$ to the portion of the boundary that touches the boundary of the half-space. Depending on $λ$, sets that minimize this capillarity perimeter among those with fixed volume are known to be suitable truncated balls lying on the boundary of the half-space.
We first give a new proof based on an ABP-type technique of the sharp isoperimetric inequality for this class of capillarity problems. Next we prove two quantitative versions of the inequality: a first sharp inequality estimates the Fraenkel asymmetry of a competitor with respect to the optimal bubbles in terms of the energy deficit; a second inequality estimates a notion of asymmetry for the part of the boundary of a competitor that touches the boundary of the half-space in terms of the energy deficit.
After a symmetrization procedure, the quantitative inequalities follow from a novel combination of a quantitative ABP method with a selection-type argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04675 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative isoperimetric inequalities for classical capillarity problems Pascale, Giulio Pozzetta, Marco Analysis of PDEs Differential Geometry Optimization and Control We consider capillarity functionals which measure the perimeter of sets contained in a Euclidean half-space assigning a constant weight $λ\in (-1,1)$ to the portion of the boundary that touches the boundary of the half-space. Depending on $λ$, sets that minimize this capillarity perimeter among those with fixed volume are known to be suitable truncated balls lying on the boundary of the half-space. We first give a new proof based on an ABP-type technique of the sharp isoperimetric inequality for this class of capillarity problems. Next we prove two quantitative versions of the inequality: a first sharp inequality estimates the Fraenkel asymmetry of a competitor with respect to the optimal bubbles in terms of the energy deficit; a second inequality estimates a notion of asymmetry for the part of the boundary of a competitor that touches the boundary of the half-space in terms of the energy deficit. After a symmetrization procedure, the quantitative inequalities follow from a novel combination of a quantitative ABP method with a selection-type argument. |
| title | Quantitative isoperimetric inequalities for classical capillarity problems |
| topic | Analysis of PDEs Differential Geometry Optimization and Control |
| url | https://arxiv.org/abs/2402.04675 |