A result on the phase diagram of a Ginzburg-Landau problem

Working with a particular modelization of Ginzburg-Landau phenomenological theory (see \cite{dutourII}, \cite{dutour} and Section \ref{change-var}), we first recall the form of the phase diagram of this modelization as it usually drawn in the physical literature (\cite{tink}, \cite{Kittel}, \cite{sarma} and \cite{PG-de-Gennes}). We then study in detail the special case, when the critical Ginzburg Landau parameter $k$ is equal to $\frac{1}{\sqrt{2}}$. This allows us to prove that the critical magnetic field $H_{c1}(k)$ is strictly decreasing at $k=\frac{1}{\sqrt{2}}$. PACS: 01.30.Cc, 02.30.Jr, 74.25.Dw