×

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.

MSC:

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
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid