Symmetry and symmetry breaking in interpolation inequalities for two-dimensional spinors -- Preliminary results

TL;DR

This paper investigates symmetry properties of optimal functions in a spinorial inequality on two-dimensional space, revealing new insights into symmetry breaking and stability through analytical and numerical methods.

Contribution

It introduces a spinorial analogue of the Caffarelli-Kohn-Nirenberg inequality, analyzes symmetry breaking, and refines the understanding of phase transitions between symmetry and asymmetry.

Findings

Numerical evidence shows symmetry breaking region extends beyond previous bounds.

Threshold of known symmetry region is linearly stable.

Refined estimates of phase transition between symmetry and symmetry breaking.

Abstract

On the two-dimensional Euclidean space, we study a spinorial analogue of the Caffarelli-Kohn-Nirenberg inequality involving weighted gradient norms. This (SCKN) inequality is equivalent to a spinorial Gagliardo-Nirenberg type interpolation inequality on a cylinder as well as to an interpolation inequality involving Aharonov-Bohm magnetic fields, which was analyzed in a paper of 2020. We examine the symmetry properties of optimal functions by linearizing the associated functional around radial minimizers. We prove that the stability of the linearized problem is equivalent to the positivity of a $2\times2$ matrix-valued differential operator. We study the positivity issue via a combination of analytical arguments and numerical computations. In particular, our results provide numerical evidence that the region of symmetry breaking extends beyond what was previously known, while the threshold of the known symmetry region is linearly stable. Altogether, we obtain refined estimates of the phase transition between symmetry and symmetry breaking. Our results also put in evidence striking differences with the three-dimensional (SCKN) inequality that was recently investigated.