SEMICLASSICAL SOBOLEV CONSTANTS FOR THE ELECTRO-MAGNETIC ROBIN LAPLACIAN

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 electro-magnetic 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.