Semiclassical Sobolev constants for the electro-magnetic Robin Laplacian. (English) Zbl 1386.35364

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 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.


35Q40 PDEs in connection with quantum mechanics
35J25 Boundary value problems for second-order elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI arXiv Euclid