The distribution of 3D superconductivity near the second critical field

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.