  We study the magnetic Laplacian and the Ginzburg-Landau functional in a thin
planar, smooth, tubular domain and with a uniform applied magnetic field. We
provide counterexamples to strong diamagnetism, and as a consequence, we prove
that the transition from the superconducting to the normal state is
non-monotone. In some non-linear regime, we determine the structure of the
order parameter and compute the super-current along the boundary of the sample.
Our results are in agreement with what was observed in the Little-Parks
experiment, for a thin cylindrical sample.
