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.
Søren FOURNAIS. Loïc LE TREUST. Nicolas RAYMOND. Jean VAN SCHAFTINGEN. "Semiclassical Sobolev constants for the electro-magnetic Robin Laplacian." J. Math. Soc. Japan 69 (4) 1667 - 1714, October, 2017. https://doi.org/10.2969/jmsj/06941667