Decay of superconductivity away from the magnetic zero set

Abstract

Abstract. We establish exponential bounds on the Ginzburg-Landau order parameter away from the curve where the applied magnetic field vanishes. In the units used in this paper, the estimates are valid when the parameter measuring the strength of the applied magnetic field is comparable with the Ginzburg-Landau parameter. This completes a previous work by the authors analyzing the case when this strength was much higher. Our results display the distribution of surface and bulk superconductivity and are valid under the assumption that the magnetic field is Hölder continuous.

On the Ginzburg–Landau energy with a magnetic field vanishing along a curve

“…• If H C 1 < H < H C 2 , the superconductor is in the mixed phase, where both the superconducting and normal states co-exist in the bulk of the sample; the most interesting aspect of the mixed phase is that the region with the normal state appears in the form of a lattice of point defects, covering the whole bulk of the sample [25] ; • If H C 2 < H < H C 3 , superconductivity disappears in the bulk but survives on the surface of the superconductor ; • If H > H C 3 , superconductivity is destroyed and the superconductor returns to the normal state . The case of a non-uniform sign changing magnetic field has been addressed first in [23] then recently in [4,5,6,17,19]. In the presence of such magnetic fields, the behavior of the superconductor (and the associated critical magnetic fields) differ significantly from the case of a uniform 1 applied magnetic field.…”

“…In particular, the order of the intensity of the third critical field H C 3 increases, and in the mixed phase between H C 2 and H C 3 , superconductivity is neither present everywhere in the bulk, nor it is evenly distributed in the form of a lattice. We refer to [17,19] for more details.…”

“…The asymptotic analysis of E gs (κ, H) has been carried for other regimes of the magnetic field strength, down to H ≈ κ 1/3 , in [4,5,19]. The case where the function B 0 is only Hölder continuous or a step function has been discussed in [17,3].…”