Quantitative stability of Sobolev inequalities on compact Riemannian manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916259694641152 |
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| author | Nobili, Francesco Parise, Davide |
| author_facet | Nobili, Francesco Parise, Davide |
| contents | We study quantitative stability results for different classes of Sobolev inequalities on general compact Riemannian manifolds. We prove that, up to constants depending on the manifold, a function that nearly saturates a critical Sobolev inequality is quantitatively $W^{1,2}$-close to a non-empty set of extremal functions, provided that the corresponding optimal Sobolev constant satisfies a suitable strict bound. The case of sub-critical Sobolev inequalities is also covered. Finally, we discuss degenerate phenomena in our quantitative controls. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_15966 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative stability of Sobolev inequalities on compact Riemannian manifolds Nobili, Francesco Parise, Davide Analysis of PDEs Differential Geometry Functional Analysis We study quantitative stability results for different classes of Sobolev inequalities on general compact Riemannian manifolds. We prove that, up to constants depending on the manifold, a function that nearly saturates a critical Sobolev inequality is quantitatively $W^{1,2}$-close to a non-empty set of extremal functions, provided that the corresponding optimal Sobolev constant satisfies a suitable strict bound. The case of sub-critical Sobolev inequalities is also covered. Finally, we discuss degenerate phenomena in our quantitative controls. |
| title | Quantitative stability of Sobolev inequalities on compact Riemannian manifolds |
| topic | Analysis of PDEs Differential Geometry Functional Analysis |
| url | https://arxiv.org/abs/2405.15966 |