Gamma-limit of the Ginzburg-Landau Energy in Thin Domain with a Large Magnetic Field

Tien-Tsan Shieh
Department of Mathematics
University of Arizona

Abstract: A one-dimensional Ginzburg-Landau model describing a superconducting closed thin wire with arbitrary cross-section subject to a large applied magnetic field is derived from the three-dimensional Ginzburg-Landau energy in the spirit of $\Gamma$-convergence. Our result proves the validity of the formal result of Richardson and Rubinstein revealing the double limit of large field and thin domain. An additional magnetic potential related to the applied field is found in the limiting functional which yields a parabolic background for the oscillatory phase transition curve between the normal and superconducting states.