SUPERCONDUCTIVITY IN DOMAINS WITH CORNERS

We study the two-dimensional Ginzburg-Landau functional in a domain with corners for exterior magnetic field strengths near the critical field where the transition from the superconducting to the normal state occurs. We discuss and clarify the definition of this field and obtain a complete asymptotic expansion for it in the large $\kappa$ regime. Furthermore, we discuss nucleation of superconductivity at the boundary.

[1]  Michael Tinkham,et al.  Introduction to Superconductivity , 1975 .

[2]  Xingbin Pan.  Upper critical field for superconductors with edges and corners , 2002 .

[3]  Daniel Phillips,et al.  The breakdown of superconductivity due to strong fields for the Ginzburg-Landau model , 1999 .

[4]  Superconductivity near critical temperature , 2003 .

[5]  Bernard Helffer,et al.  Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 , 2001 .

[6]  Xing-Bin Pan,et al.  Gauge Invariant Eigenvalue Problems in ℝⁿ and in ℝⁿ , 1999 .

[7]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[8]  Shmuel Agmon,et al.  Lectures on exponential decay of solutions of second order elliptic equations : bounds on eigenfunctions of N-body Schrödinger operators , 1983 .

[9]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[10]  M. Dauge,et al.  Asymptotics for the Low-Lying Eigenstates of the Schrödinger Operator with Magnetic Field near Corners , 2006 .

[11]  Xing-Bin Pan,et al.  Surface superconductivity in 3 dimensions , 2004 .

[12]  Andrew J. Bernoff,et al.  Onset of superconductivity in decreasing fields for general domains , 1998 .

[13]  Bernard Helffer,et al.  Strong diamagnetism for general domains and application , 2007 .

[14]  B. Helffer,et al.  Magnetic Bottles in Connection with Superconductivity , 2001 .

[15]  Bernard Helffer,et al.  Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian , 2006 .

[16]  D. R. Tilley,et al.  Superfluidity and Superconductivity , 2019 .

[17]  Xing-Bin Pan,et al.  Eigenvalue problems of Ginzburg–Landau operator in bounded domains , 1999 .

[18]  P. Sternberg,et al.  Boundary Concentration for Eigenvalue Problems Related to the Onset of Superconductivity , 2000 .

[19]  Xing-Bin Pan,et al.  Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity , 1999 .

[20]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[21]  Bernard Helffer,et al.  Upper critical field and location of surface nucleation of superconductivity , 2003 .

[22]  François Alouges,et al.  Numerical computations of fundamental eigenstates for the Schrödinger operator under constant magnetic field , 2006 .

[23]  Xing-Bin Pan,et al.  Surface Nucleation of Superconductivity in 3-Dimensions , 2000 .

[24]  Qiang Du,et al.  Analysis and Approximation of the Ginzburg-Landau Model of Superconductivity , 1992, SIAM Rev..

[25]  Virginie Bonnaillie.  On the fundamental state energy for a Schrödinger operator with magnetic field in domains with corners , 2005 .

[26]  Virginie Bonnaillie.  On the fundamental state for a Schrödinger operator with magnetic field in a domain with corners , 2003 .

[27]  B. Helffer,et al.  Strong diamagnetism for general domains and applications , 2006, math-ph/0607071.

[28]  H. Jadallah.  The onset of superconductivity in a domain with a corner , 2001 .

[29]  G. Vial,et al.  Computations of the first eigenpairs for the Schrödinger operator with magnetic field , 2007 .

[30]  B. Helffer,et al.  On the Third Critical Field in Ginzburg-Landau Theory , 2006 .

[31]  P. Silverman,et al.  Type II superconductivity , 1969 .