Quantitative stability of Sobolev inequalities on compact Riemannian manifolds

Francesco Nobili; Davide Parise

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arXiv · 2405.15966

Quantitative stability of Sobolev inequalities on compact Riemannian manifolds

Francesco Nobili·Davide Parise

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Abstract

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.

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Francesco Nobili, Davide Parise. 2024-05-24. Quantitative stability of Sobolev inequalities on compact Riemannian manifolds. https://arxiv.org/abs/2405.15966

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