arXiv subjects
Publications and source records attributed to Sascha Eichmann.
We are looking for an optimal convex domain on which the boundary value problem $$\left\{\begin{array}{cc}(-\Delta)^2 u_\gamma-\gamma\Delta u_\gamma = f,& \mbox{ in }\Omega\\ u_\gamma=\partial_\nu u_\gamma=0,& \mbox{ on }\partial\Omega\end{array}\right.$$ admits a nonnegative solution for the most $\gamma$, if $f$ is a given nonnegative function.
To avoid possible singularities in the Willmore flow, one usually works under an energy threshold provided by the Li-Yau inequality. Here we improve this threshold by also considering parts outside of a possible singularity together with Dirichlet boundary data. We work in the class of surfaces of revolution.
In this article we consider positivity issues for the clamped plate equation with high tension $γ>0$. This equation is given by $Δ^2u - γΔu=f$ under clamped boundary conditions. Here we show, that given a positive $f$, i.e. upwards pushing, we find a $γ_0>0$ such that for all $γ\geq γ_0$ the bending $u$ is indeed positive. This $γ_0$ only depends on the domain and the ratio of the $L^1$ and $L^\infty$ norm of $f$. In contrast to a recent result by Cassani&Tarsia, our approach is valid in all dimensions.
We examine the phase dependent Helfrich energy and show an Euler-Lagrange equation on the phase seperation line. This result has already been observed by e.g. Jülicher-Lipowski and later Elliot-Stinner. Here we are able to lower the regularity assumption for this result down to $C^{1,1}$ for the seperation line. In the proof we employ a carefully choosen test function utilising the signed distance function.
We minimise the Canham-Helfrich energy in the class of closed immersions with prescribed genus, surface area and enclosed volume. Compactness is achieved in the class of oriented varifolds. The main result is a lower-semicontinuity estimate for the minimising sequence, which is in general false by a counter example by Große-Brauckmann. The main argument involved is showing partial regularity of the limit. It entails comparing the Helfrich energy of the minimising sequence locally to that of a biharmonic graph. This idea is by Simon, but it cannot be directly applied, since the area and enclosed volume of the graph may differ. By an idea of Schygulla we adjust these quantities by using a two parameter diffeomorphism of $\mathbb{R}^3$
We construct a branched Helfrich immersion satisfying Dirichlet boundary conditions. The number of branch points is finite. We proceed by a variational argument and hence examine the Helfrich energy for oriented varifolds. The main contribution of this paper is a lower-semicontinuity result with respect to oriented varifold convergence for the Helfrich energy and a minimising sequence. For arbitrary sequences this is false by a counterexample of Große-Brauckmann.