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 $L^\infty$-norm of the corresponding minimizers near the magnetic wells. 
Soeren Fournais and Nicolas Raymond
math.AP