  The aim of this paper is to establish estimates of the lowest eigenvalue of
the Neumann realization of $(i\nabla+B\textbf{A})^2$ on an open bounded subset
of $\mathbb{R}^2$ $\Omega$ with smooth boundary as $B$ tends to infinity. We
introduce a "magnetic" curvature mixing the curvature of $\partial\Omega$ and
the normal derivative of the magnetic field and obtain an estimate analogous
with the one of constant case. Actually, we give a precise estimate of the
lowest eigenvalue in the case where the restriction of magnetic field to the
boundary admits a unique minimum which is non degenerate. We also give an
estimate of the third critical field in Ginzburg-Landau theory in the variable
magnetic field case.
