Interpolation inequalities on the sphere and phase transition: rigidity, symmetry and symmetry breaking

TL;DR

This paper investigates phase transitions in Gagliardo-Nirenberg-Sobolev inequalities on the sphere, analyzing symmetry, symmetry breaking, and the nature of phase transitions through solution properties.

Contribution

It characterizes symmetry and symmetry-breaking regimes and identifies conditions for first and second order phase transitions in these inequalities.

Findings

Identifies regimes of symmetry and symmetry breaking.

Establishes conditions for first and second order phase transitions.

Analyzes solution branches of Euler-Lagrange equations.

Abstract

This paper is devoted to the study of phase transitions associated to a large family of Gagliardo-Nirenberg-Sobolev interpolation inequalities on the sphere depending on one parameter. We characterize symmetry and symmetry breaking regimes, with a phase transition that can be of first or second order. We establish various new results and study the qualitative properties of the branches of solutions to the Euler-Lagrange equations.