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 ($C\lambda^6$).
Our methods, which build on the work of Lin, Garofalo, and Han, can also be used to study solutions to other problems. We discuss some of these possibilities.

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.
Due to this observation and based on numerical experiments, we expect that this flow develops singularities in finite or infinite time in general. We will show that the situation changes dramatically if one looks at planar curves: We will see that for planar curves we have smooth existence for all times and convergence to circles as time goes to infinity.
The basic tool in the proof is the construction of blowup profiles at possible singularities which also works for arbitrary codimension. These satisfy a type of pseudo-differential equation which in codimension one does not possess a non-trivial solution.