Optimal magnetic Sobolev constants in the semiclassical limit

TL;DR

This paper investigates the behavior of optimal magnetic Sobolev constants in a bounded domain as the semiclassical parameter approaches zero, providing explicit estimates and analyzing the localization of minimizers near magnetic wells.

Contribution

It offers the first quantitative estimates of magnetic Sobolev constants in the semiclassical limit with explicit dependence on the parameter and studies minimizer localization.

Findings

Explicit semiclassical estimates of magnetic Sobolev constants.

Demonstration of exponential localization of minimizers near magnetic wells.

Analysis of the dependence of constants on the semiclassical parameter.

Abstract

This paper is devoted to the semiclassical analysis of the best constants in the magnetic Sobolev embeddings in the case of a bounded domain of the plane carrying Dirichlet conditions. We provide quantitative estimates of these constants (with an explicit dependence on the semiclassical parameter) and analyze the exponential localization in $L^\infty$-norm of the corresponding minimizers near the magnetic wells.