Calculus of Variations and Geometric Measure Theory

F. Nobili - D. Parise

Quantitative stability of Sobolev inequalities on compact Riemannian manifolds

created by nobili on 28 May 2024



Inserted: 28 may 2024

Year: 2024

ArXiv: 2405.15966 PDF


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.