Superconductivity in mesoscopic domains

Let M be a graph embedded in the plane. The lace around M is the open set O_ε of thickness 2ε around the edges of M, and smoothed close to the vertices of M. The details of the smoothing will be proved to be irrelevant. The unknowns in the two-dimensional Ginzburg-Landau functional are the vector potential and the complex order parameter. As ε tends to 0, it is possible to extract from any sequence of minimizers a converging subsequence whose limit is a minimizer of a one-dimensional Ginzburg-Landau functional on M. The only unknown in this functional is a complex order parameter. A further reduction leads to a new functional depending only on a realvalued function and n integers, n being the number of independent cycles of the graph, or equivalently the number of holes in R² \ M.