Mathematics

We derive estimates for the principal eigenvalue $\lambda^{\Omega}_1(\alpha)$ corresponding to $\Delta u = \lambda(\alpha) u $ in $\Omega$, $\frac {\partial u}{\partial \nu} = \alpha \, u$ on $\partial \Omega$, with $\Omega \in {\cal R^n}$ a $C^{0,1}$ bounded domain, and $\alpha$ a fixed real. If $\alpha0$, we have $\lambda^{\Omega}_1(\alpha)>0$ and the eigenvalue problem arises in reaction-diffusion equations, as well as enhanced surface superconductivity. In the context of superconductivity, our results show the increase of the critical temperature in zero fields for systems with enhanced surface superconductivity.
This is joint work with Dr. R. Smits.