Xing-Bin Pan

Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity

Abstract In this paper we study the effects of an applied magnetic field on a superconductor and estimate the value of the upper critical magnetic field HC3 at which superconductivity can nucleate. In the case of a spatially homogeneous applied field, we show that HC3≃κ/β0, the ratio of the Ginzburg–Landau parameter κ and the first eigenvalue β0 of a twisted Laplacian operator, and that superconductivity nucleates at the boundary when the applied field is close to HC3. In the case of a spatially non-homogeneous applied field, we give an estimate for the upper critical value and find that superconducting properties may persist only in the interior of the domain. In addition, we show that the order parameter concentrates at the minimum points of the applied field.

Eigenvalue problems of Ginzburg–Landau operator in bounded domains

In this paper we study the eigenvalue problems for the Ginzburg–Landau operator with a large parameter in bounded domains in R2 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.

Upper critical field and location of surface nucleation of superconductivity

Abstract In this paper we improve the estimate obtained by Lu–Pan on the value of the upper critical field HC3(κ) for a cylindrical superconductor with cross section Ω being an arbitrary 2-dimensional smooth bounded domain. We also show that, when a homogeneous magnetic field is applied along the axis of the cylinder with magnitude of the field close to HC3, superconductivity nucleates first at the surface of the sample where the curvature of ∂ Ω is maximal.

Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains

We establish an asymptotic estimate of the lowest eigenvalue μ(bF) of the Schrodinger operator -⊇ 2 bF with a magnetic field in a bounded 2-dimensional domain, where curl F vanishes non-degenerately, and b is a large parameter. Our study is based on an analysis on an eigenvalue variation problem for the Sturm-Liouville problem. Using the estimate, we determine the value of the upper critical field for superconductors subject to non-homogeneous applied magnetic fields, and localize the nucleation of superconductivity.

Positive solutions of super-critical elliptic equations and asymptotics

This paper is devoted to the study of positive solutions of semilinear elliptic equations . Asymp- totic behavior of ground states and uniqueness of singular ground states are proved via invariant manifold theory of dynamical systems. The Dirichlet problem in exterior domains is also studied. It is proved that this problem has one positive solution with fast decay and infinitely many positive solutions with slow decay. The asymptotics of the singular sequence of fast decay solutions when p approaches to is also discussed.

Surface superconductivity in 3 dimensions

We study the Ginzburg-Landau system for a superconductor occupying a 3-dimensional bounded domain, and improve the estimate of the upper critical field H C3 obtained by K. Lu and X. Pan in J. Diff. Eqns., 168 (2000). 386-452. We also analyze the behavior of the order parameters. We show that, under an applied magnetic field lying below and not far from H C3 order parameters concentrate in a vicinity of a sheath of the surface that is tangential to the applied field, and exponentially decay both in the normal and tangential directions away from the sheath in the L 2 sense. As the applied field decreases further but keeps in between and away from H C2 and H C3 , the superconducting sheath expands but does not cover the entire surface, and superconductivity at the surface portion orthogonal to the applied field is always very weak. This phenomenon is significantly different to the surface superconductivity on a cylinder of infinite height studied by X. Pan in Comm. Math. Phys., 228 (2002), 327-370, where under an axial applied field lying in-between H C2 and H C3 the entire surface is in the superconducting state.

Magnetic field-induced instabilities in liquid crystals

We use the Landau–de Gennes model to investigate the magnetic field‐induced instabilities in liquid crystals. In particular, we examine the change of weak and strong stabilities in the pure smectic states and in the pure nematic states. Motivated by de Gennes’ discovery on the analogies between liquid crystals and superconductors, we introduce critical magnetic fields

On an elliptic equation related to the blow-up phenomenon in the nonlinear Schrödinger equation

H_{\text{\rm s}}

Superconductivity near the normal state in a half-plane under the action of a perpendicular electric current and an induced magnetic field

and

Superconductivity near critical temperature

H_{\rm sh}