About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities

TL;DR

This paper reviews recent findings on the existence, symmetry, and symmetry breaking of extremal functions for certain interpolation inequalities, highlighting conditions under which symmetry is lost and analyzing critical cases.

Contribution

It synthesizes recent results on symmetry breaking phenomena for Caffarelli-Kohn-Nirenberg and weighted logarithmic Hardy inequalities, emphasizing stability and critical cases.

Findings

Symmetry breaking occurs where symmetric extremals are linearly stable.

Extremals are not radially symmetric in certain parameter regions.

Critical cases for CKN and WLH inequalities are thoroughly analyzed.

Abstract

This article is devoted to a review of some recent results on existence, symmetry and symmetry breaking of optimal functions for Caffarelli-Kohn-Nirenberg (CKN) and weighted logarithmic Hardy (WLH) inequalities. These results have been obtained in a series of papers in collaboration with M. del Pino, S. Filippas, M. Loss, G. Tarantello and A. Tertikas. Here we put the highlights on a symmetry breaking result: extremals of some inequalities are not radially symmetric in regions where the symmetric extremals are linearly stable. Special attention is paid to the study of the critical cases for (CKN) and (WLH).