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.
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