Katarína Bellová and Simon Blatt
16:15 Uhr | Katarína Bellová (MPI Leipzig) | Nodal sets of Steklov eigenfunctions
Abstract
We study the nodal set of the Steklov eigenfunctions on the boundary of
a smooth bounded domain in $\mathbb R^n$
- the eigenfunctions of the Dirichlet-to-Neumann map. For a bounded
Lipschitz domain $\Omega\subset\mathbb R^n$,
this map associates to each function $u$ defined on the boundary
$\partial \Omega$, the normal derivative
of the harmonic function on $\Omega$ with boundary data $u$.
Under the assumption that the domain $\Omega$ is $C^2$, we prove a
doubling property for the eigenfunction $u$.
We estimate the Hausdorff $\mathcal H^{n-2}$-measure of the nodal set of
$u$ in terms of the eigenvalue
$\lambda$ as $\lambda$ grows to infinity, provided $\Omega$ is fixed. In
case that the domain $\Omega$ is analytic,
we prove a polynomial bound ($C\lambda^6$).
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17:45 Uhr | Simon Blatt (Karlsruhe) | The gradient flow of the Möbius energy
In this talk I will present some recent results regarding the gradient flow of the Möbius energy introduced by Jun O'Hara in 1991.
Modeled to punish self-intersections of curves, one of the most striking feature of this energy is that it is invariant under the group of
Möbius transformations - a feature it has in common with the Willmore energy.
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