Stability for the Sobolev inequality in cones

TL;DR

This paper establishes a quantitative stability version of the Sobolev inequality in cones, demonstrating that nondegeneracy of minimizers ensures stability, with more precise results when classical bubbles are minimizers.

Contribution

It proves a new stability result for the Sobolev inequality in cones, extending previous work to a broader class of cones and analyzing the role of minimizer degeneracy.

Findings

Stability holds for cones with nondegenerate minimizers.

More precise stability results are obtained for classical bubble minimizers.

Local estimates are insufficient for optimal constants in the inequality.

Abstract

We prove a quantitative Sobolev inequality in cones of Bianchi-Egnell type, which implies a stability property. Our result holds for any cone as long as the minimizers of the Sobolev quotient are nondegenerate, which is the case of most cones. When the minimizers are the classical bubbles we have more precise results. Finally, we show that local estimates are not enough to get the optimal constant for the quantitative Sobolev inequality.