Optimal magnetic Sobolev constants in the semiclassical limit

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 $L^{\mathrm{\infty }}$-norm of the corresponding minimizers near the magnetic wells.