The distribution of 3D superconductivity near the second critical field

Abstract

Abstract. We study the minimizers of the Ginzburg-Landau energy functional with a constant magnetic field in a three dimensional bounded domain. The functional depends on two positive parameters, the Ginzburg-Landau parameter and the intensity of the applied magnetic field, and acts on complex valued functions and vector fields. We establish a formula for the distribution of the L 2 -norm of the minimizing complex valued function (order parameter). The formula is valid in the regime where the Ginzburg-Landau parameter is large and the applied magnetic field is close to the second critical field-the threshold value corresponding to the transition from the superconducting to the normal phase in the bulk of the sample. Earlier results are valid in 2D domains and for the L 4 -norm in 3D domains.

The density of superconductivity in the bulk regime

“…This result is complementary to the results in [14] and [19] devoted respectively to the regimes b > 1 (surface superconductivity) and b → 1 − (bulk superconductivity near H C 2 ) for three dimensional superconducting samples.…”

“…This problem is related to the one of the Abrikosov state near the critical field H C 2 := κ , where the transition to the normal state in the bulk occurs. This is visualized in the regime b → 1 − in (1.7) and is analyzed in many papers, [3,12,18,19]. The same difficulty is encountered when trying to detect the vortices by the methods of Sandier-Serfaty, so that the analysis is shifted to the distribution of the density |ψ| 2 instead.…”

Decay of superconductivity away from the magnetic zero set

“…The proof is reminiscent of the method used by the second author in [26,Sec. 4] (see also [27]). We assume that B 0 (x 0 ) > 0.…”