Attainability of the best Sobolev constant in a ball

TL;DR

This paper investigates a scale-invariant form of the Sobolev inequality within a ball and demonstrates that its optimal constant is achieved by Aubin-Talenti type functions, extending to Caffarelli-Kohn-Nirenberg inequalities.

Contribution

It introduces a new scale-invariant Sobolev inequality in a bounded domain and proves the attainability of its best constant by Aubin-Talenti functions, extending known results.

Findings

Best constant in the new inequality is attained by Aubin-Talenti functions.

Extension of results to Caffarelli-Kohn-Nirenberg inequalities.

Establishes attainability despite lack of dilation invariance in bounded domains.

Abstract

The best constant of the Sobolev inequality in the whole space is attained by the Aubin-Talenti function; however, this does not happen in bounded domains because the break in dilation invariance. In this paper, we investigate a new scale invariant form of the Sobolev inequality in a ball and show that its best constant is attained by functions of the Aubin-Talenti type. Generalization to the Caffarelli-Kohn-Nirenberg inequality in a ball is also discussed.