  We prove that the lowest eigenvalue of the Laplace operator with a magnetic
field having a self-intersecting zero set is a monotone function of the
parameter defining the strength of the magnetic field, in a neighborhood of
infinity. We apply this monotonicity result on the study of the transition from
superconducting to normal states for the Ginzburg-Landau model, and prove that
the transition occurs at a unique threshold value of the applied magnetic
field.
