Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces

TL;DR

This paper investigates symmetry breaking in nonlinear flows on cylinders and Euclidean spaces, providing sharp constants, solving a longstanding conjecture, and establishing new eigenvalue estimates on non-compact manifolds.

Contribution

It introduces a novel method using generalized entropy functionals to analyze symmetry breaking, solving a key conjecture and deriving new eigenvalue bounds.

Findings

Solved a longstanding conjecture on symmetry range.

Derived sharp estimates for Schrödinger operator eigenvalues.

Established a new approach for functional inequalities on non-compact manifolds.

Abstract

This paper is motivated by the characterization of the optimal symmetry breaking region in Caffarelli-Kohn-Nirenberg inequalities. As a consequence, optimal functions and sharp constants are computed in the symmetry region. The result solves a longstanding conjecture on the optimal symmetry range. As a byproduct of our method we obtain sharp estimates for the principal eigenvalue of Schr\"odinger operators on some non-flat non-compact manifolds, which to the best of our knowledge are new. The method relies on generalized entropy functionals for nonlinear diffusion equations. It opens a new area of research for approaches related to carr\'e du champ methods on non-compact manifolds. However key estimates depend as much on curvature properties as on purely nonlinear effects. The method is well adapted to functional inequalities involving simple weights and also applies to general cylinders. Beyond results on symmetry and symmetry breaking, and on optimal constants in functional inequalities, rigidity theorems for nonlinear elliptic equations can be deduced in rather general settings.