We introduce the following nonlinear eigenvalue, or optimal magnetic Sobolev constant:

$\newcommand{\indice}[2]{_}$

$$\lambda(\Omega, {\bf A}, p,h)=\inf_{\psi\in H^1_{0}(\Omega), \psi\neq 0}\frac{\indice{\mathcal{Q}}{h,{\bf A}}(\psi)} {\left(\int_{\Omega}|\psi|^p dx \right)^{\frac{2}{p}}}=\inf_{\underset{ |\psi|{ L^p(\Omega)}=1}{\psi\in H^1{0} (\Omega),}}\mathcal{Q}_{h,{\bf A}}(\psi), $$

where the magnetic quadratic form is defined by

$\forall \psi\in H^1_{0}(\Omega),\quad\indice{\mathcal{Q}}{h,{\bf A}}(\psi)=\int_{\Omega}|(-ih\nabla+{\bf A})\psi|^2 dx.$

This object, and the corresponding minimizing functions, are of obvious interest in non-linear evolution problems.

We obtain---under different classes of assumptions on the magnetic field generated by the vector potential ${\bf A}$---leading order asymptotic estimates on $\lambda(\Omega, {\bf A}, p,h)$ as well as localisation estimates for the minimizers.

This work is based on collaboration with Nicolas Raymond.

Slides