Semiclassical Sobolev constants for the electro-magnetic Robin Laplacian

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.
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Soeren Fournais, Loïc Le Treust, Nicolas Raymond, Jean van Schaftingen. Semiclassical Sobolev constants for the electro-magnetic Robin Laplacian. Journal of the Mathematical Society of Japan, Maruzen Company Ltd, 2017, 69 (4), pp.1667-1714. ⟨10.2969/jmsj/06941667⟩. ⟨hal-01285311⟩

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