Superconductivity in mesoscopic domains

Let M be a graph embedded in the plane. The lace around M is

the open

set O of thickness

2

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

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 real

valued function and n integers, n being the number of indepen

dent

cycles of the graph, or equivalently the number of holes in

R² \ M.