Boundary concentration for eigenvalue problems related to the onset of superconductivity. (English) Zbl 0982.35077

del Pino, Manuel; Felmer, Patricio L.; Sternberg, Peter

Commun. Math. Phys. 210, No. 2, 413-446 (2000).

The authors deal with the asymptotic behaviour of the eigenvalue \(\mu(h)\) and corresponding eigenfunction associated with the variational problem \[ \mu(h)\equiv \inf_{\psi\in H^1(\Omega, \mathbb{C})} {\int_\Omega|(i\nabla+ hA)\psi|^2 dx dy\over \int_\Omega|\psi|^2 dx dy} \] in the regime \(h\gg 1\). Here, \(A\) is any vector field with curl equal to \(1\). The authors show that when the domain \(\Omega\) is not a disc, the first eigenfunction does not concentrate along the entire boundary. It must be decay to zero with large \(h\) somewhere along the boundary, while simultaneously decaying at an exponential rate inside the domain.

Reviewer: Messoud Efendiev (Berlin)

MSC:

35P15	Estimates of eigenvalues in context of PDEs
35J10	Schrödinger operator, Schrödinger equation
35Q60	PDEs in connection with optics and electromagnetic theory
82D55	Statistical mechanics of superconductors

Keywords:

eigenvalue; eigenfunction; variational problem; exponential rate