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\noindent{\bf Mathematisches Oberseminar} {\it PDG und
Spektraltheorie} (WiSe 2014/15).

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\noindent{\bf Date:} 06.11.2014.

\noindent{\bf Time and place:} 14:15 in B 134.

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\noindent{\bf Speaker:} S{\o}ren Fournais (Aarhus University).

\noindent{\bf Titel:} {\it Optimal magnetic Sobolev constants in the semiclassical limit.}


\noindent{\bf Abstract:}

\noindent We introduce the following nonlinear eigenvalue, or optimal
magnetic Sobolev constant: 
$$
  \lambda(\Omega, {\bf A}, p,h)=\inf_{\psi\in H^1_{0}(\Omega), \psi\neq
  0}\frac{\mathcal{Q}_{h,{\bf A}}(\psi)}{\left(\int_{\Omega}|\psi|^p dx
  \right)^{\frac{2}{p}}}=\inf_{\underset{ \|\psi\|_{
  L^p(\Omega)}=1}{\psi\in H^1_{0}(\Omega),}}\mathcal{Q}_{h,{\bf
  A}}(\psi), 
$$
where the magnetic quadratic form is defined by
$$\forall \psi\in H^1_{0}(\Omega),\quad\mathcal{Q}_{h,{\bf
A}}(\psi)=\int_{\Omega}|(-ih\nabla+{\bf A})\psi|^2 dx.$$ 

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

\noindent 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.  

\noindent This work is based on collaboration with Nicolas Raymond (Rennes).

\noindent

\noindent 

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Thomas {\O}stergaard S{\o}rensen

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