Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909560454774784 |
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| author | Dolbeault, Jean Esteban, Maria J. Figalli, Alessio Frank, Rupert L. Loss, Michael |
| author_facet | Dolbeault, Jean Esteban, Maria J. Figalli, Alessio Frank, Rupert L. Loss, Michael |
| contents | We prove a sharp quantitative version for the stability of the Sobolev inequality with explicit constants. Moreover, the constants have the correct behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative stability estimate for the Gaussian log-Sobolev inequality with an explicit dimension-free constant. Our proofs rely on several ingredients such as competing symmetries, a flow based on continuous Steiner symmetrization that interpolates continuously between a function and its symmetric decreasing rearrangement, and refined estimates on the Sobolev functional in the neighborhood of the optimal Aubin--Talenti functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_08651 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence Dolbeault, Jean Esteban, Maria J. Figalli, Alessio Frank, Rupert L. Loss, Michael Analysis of PDEs Classical Analysis and ODEs Functional Analysis 49J40, 26D10, 35A23 We prove a sharp quantitative version for the stability of the Sobolev inequality with explicit constants. Moreover, the constants have the correct behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative stability estimate for the Gaussian log-Sobolev inequality with an explicit dimension-free constant. Our proofs rely on several ingredients such as competing symmetries, a flow based on continuous Steiner symmetrization that interpolates continuously between a function and its symmetric decreasing rearrangement, and refined estimates on the Sobolev functional in the neighborhood of the optimal Aubin--Talenti functions. |
| title | Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis 49J40, 26D10, 35A23 |
| url | https://arxiv.org/abs/2209.08651 |