arXiv subjects

Nils Dabrock

Publications and source records attributed to Nils Dabrock.

3 recordsLinked to original sources

arXiv2022

"Gradient-free" diffuse approximations of the Willmore functional and Willmore flow

We introduce new diffuse approximations of the Willmore functional and the Willmore flow. They are based on a corresponding approximation of the perimeter that has been studied by Amstutz-van Goethem [{\em Interfaces Free Bound. 14 (2012)}]. We identify the candidate for the $Γ$--convergence, prove the $Γ$--limsup statement and justify the convergence to the Willmore flow by an asymptotic expansion. Furthermore, we present numerical simulations that are based on the new approximation.

math.AP↗

arXiv2019

Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs

We are concerned with a stochastic mean curvature flow of graphs over a periodic domain of any space dimension. We establish existence of martingale solutions which are strong in the PDE sense and study their large-time behavior. Our analysis is based on a viscous approximation and new global bounds, namely, an $L^{\infty}_{ω,x,t}$ estimate for the gradient and an $L^{2}_{ω,x,t}$ bound for the Hessian. The proof makes essential use of the delicate interplay between the deterministic mean curvature part and the stochastic perturbation, which permits to show that certain gradient-dependent energies are supermartingales. Our energy bounds in particular imply that solutions become asymptotically spatially homogeneous and approach a Brownian motion perturbed by a random constant.

math.PR↗

arXiv2017

Characterization of minimizers of an anisotropic variant of the Rudin-Osher-Fatemi functional with $L^1$ fidelity term

In this paper we study an anisotropic variant of the Rudin-Osher-Fatemi functional with $L^1$ fidelity term of the form \[ E(u) = \int_{\mathbb{R}^n} ϕ(\nabla u) + λ\| u -f \|_{L^1(\mathbb{R}^n)}. \] We will characterize the minimizers of $E$ in terms of the Wulff shape of $ϕ$ and the dual anisotropy. In particular we will calculate the subdifferential of $E$. We will apply this characterization to the special case $ϕ= |\cdot|_1$ and $n=2$, which has been used in the denoising of 2D bar codes. In this case, we determine the shape of a minimizer $u$ when $f$ is the characteristic function of a circle.

math.AP↗