Semiclassical Sobolev constants for the electro-magnetic Robin Laplacian (Q1687584)

This article concerns the behavior and asymptotic analysis of Sobolev constants in the semiclassical limit and intends to extend the work [the first and third author, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 33, No. 5, 1199--1222 (2016; Zbl 1350.35006)]. Here, \(d\geq 2\), and \(\Omega\subset\mathbb R^d\) with smooth boundary prescribed with a Robin boundary condition. Analysis is done in the presence of an electromagnetic potential \((V,\mathbf A)\in C^\infty(\overline{\Omega},\mathbb R\times\mathbb R^d)\) and a Robin coefficient function satisfying \(\gamma\in C^\infty(\partial\Omega,\mathbb R)\). The crucial assumption (Assumption 1.5) provides a semi-continuity property which is needed to estimate the Sobolev constants from above (see concentration-compactness arguments in [the first author and \textit{B. Helffer}, Spectral methods in surface superconductivity. Basel: Birkhäuser (2010; Zbl 1256.35001)]). The main result (Theorem 1.9) provides an exponential decay estimate for (suitable normalized) minimizers \(\psi=\psi_h\) of the associated nonlinear focusing equation, \[ \begin{cases} (-ih\nabla+\mathbf A)^2\psi+hV\psi=\lambda(h)| \psi|^{p-2}\psi, \\ (-ih\nabla+\mathbf A)\psi\cdot\mathbf n=-ih^{1/2}c\psi,\text{ on }\partial\Omega.\end{cases} \] away from the minimizers of the concentration function. Some further results concerning applications and extensions are also discussed.