On the Ginzburg-Landau critical field in three dimensions

We study the three dimensional Ginzburg-Landau model of superconductivity. Several `natural' definitions of the (third) critical field, $H_{C_3}$, governing the transition from the superconducting state to the normal state, are considered. We analyze the relation between these fields and give conditions as to when they coincide. An interesting part of the analysis is the study of the monotonicity of the ground state energy of the Laplacian, with constant magnetic field and with Neumann (magnetic) boundary condition, in a domain $\Omega$. It is proved that the ground state energy is a strictly increasing function of the field strength for sufficiently large fields. As a consequence of our analysis we give an affirmative answer to a conjecture by Pan.