Eigenvalue problems of Ginzburg–Landau operator in bounded domains

Abstract

A remark on the dimension of the attractor for the Dirichlet problem of the complex Ginzburg-Landau equationIn this paper we study the eigenvalue problems for the Ginzburg-Landau operator with a large parameter in bounded domains in R 2 under gauge invariant boundary conditions. The estimates for the eigenvalues are obtained and the asymptotic behavior of the associated eigenfunctions is discussed. These results play a key role in estimating the critical magnetic field in the mathematical theory of superconductivity.

Estimates for the lowest eigenvalue of magnetic Laplacians

“…A lot of attention has been paid to the asymptotic behavior of µ 1 (Ω, B) for large values of the magnetic field, see e.g. [Bo,FH1,LP,Ra,Si].…”

On the Third Critical Field in Ginzburg-Landau Theory

“…Our main motivation comes from superconductivity. As appeared from the works of Bernoff-Sternberg [BeSt], LuPa2,LuPa3,LuPa4], and HelfferPan [HePa], the determination of the lowest eigenvalues of the magnetic Schrödinger operator is crucial for a detailed description of the nucleation of superconductivity (on the boundary) for superconductors of Type II and for accurate estimates of the critical field H C3 . If the determination of the complete asymptotics of the lowest eigenvalues of the Schrödinger operators was essentially achieved (except for exponentially small effects) in the two-dimensional case with the works of [HeMo2] and [FoHe2], what remained to be determined was the corresponding asymptotics for the critical field.…”

Optimal uniform elliptic estimates for the Ginzburg-Landau system

“…This formula appears in [LuPa2] with an additional boundary term that we are able to show to be zero in the case of a (magnetic) Neumann-condition.…”

“…In this and the following subsections we will use the non-existence results from Subsection 4.1 to obtain improved versions of the estimates in Theorem 3.1 in a reduced parameter range. The application of this idea ('blow-up') to the GinzburgLandau system appeared to our knowledge first in [LuPa1,LuPa2] and has since been used extensively since (see for instance [LuPa4,Pan4,HePa]). …”

“…In the literature such estimates are found in varying generality scattered over different publications (c.f. [LuPa1,LuPa2,LuPa3,LuPa4], [HePa],...).…”