Semiclassical Sobolev constants for the electro-magnetic Robin Laplacian

TL;DR

This paper analyzes the asymptotic behavior of Sobolev constants for the electromagnetic Robin Laplacian in the semiclassical limit, using advanced mathematical techniques to derive localization estimates and understand the influence of electromagnetic fields and boundary conditions.

Contribution

It introduces a novel combination of semiclassical and concentration-compactness methods to study Sobolev constants with electromagnetic and Robin boundary conditions in any dimension.

Findings

Derived asymptotic formulas for Sobolev constants in the semiclassical limit.

Established exponential localization estimates for minimizers.

Extended analysis to include electromagnetic fields and Robin boundary conditions.

Abstract

This paper is devoted to the asymptotic analysis of the optimal Sobolev constants in the semiclassical limit and in any dimension. We combine semiclassical arguments and concentration-compactness estimates to tackle the case when an electromagnetic field is added as well as a smooth boundary carrying a Robin condition. As a byproduct of the semiclassical strategy, we also get exponentially weighted localization estimates of the minimizers.