arXiv subjects

Michael Gößwein

Publications and source records attributed to Michael Gößwein.

4 recordsLinked to original sources

arXiv2023

Stability analysis for the anisotropic curve shortening flow of planar networks

In this article we study the anisotropic curve shortening flow for a planar network of three curves with fixed endpoints and which meet in a triple junction. We show that the anisotropic curvature energy fulfills a Lojasiewicz-Simon gradient inequality and use this knowledge to derive stability results for the flow. Precisely, in our main theorem we show that for any initial data, which are $C^{2,α}$-close to a (local) energy minimizer, the flow exists globally and converges to a possibly different energy minimum.

math.AP↗

arXiv2020

Existence and Uniqueness of the Motion by Curvature of regular networks

We prove existence and uniqueness of the motion by curvatureof networks in $\mathbb{R}^n$ when the initial datum is of class $W^{2-\frac{2}{p}}_p$, with triple junction where the unit tangent vectors to the concurring curves form angles of $120$ degrees. Moreover we investigated the regularization effect due to the parabolic nature of the system. An application of this wellposedness result is a new proof of Theorem 3.18 in "Motion by Curvature of Planar Networks" by Mantegazza-Novaga-Tortorelli where the possible behaviors of the solutions at the maximal time of existence are described. Our study is motivated by an open question proposed in "Evolution of Networks with Multiple Junctions " by Mantegazza-Novaga-Pluda-Schulze: does there exist a unique solution of the motion by curvature of networks with initial datum a regular network of class $C^2$? We give a positive answer.

math.AP↗

arXiv2019

On the Surface Diffusion Flow with Triple Junctions in Higher Space Dimensions

We show short time existence for the evolution of triple junction clusters driven by the surface diffusion flow. On the triple line we use the boundary conditions derived by Garcke and Novick-Cohen as the singular limit of a Cahn-Hilliard equation with degenerated mobility. These conditions are concurrency of the triple junction, angle conditions between the hypersurfaces, continuity of the chemical potentials and a flux-balance. For the existence analysis we first write the geometric problem over a fixed reference surface and then use for the resulting analytic problem an approach in a parabolic Hölder setting.

math.AP↗

arXiv2019

Non-linear Stability of Double Bubbles under Surface Diffusion

We consider the evolution of triple junction clusters driven by the surface diffusion flow. On the triple line we use the boundary conditions derived by Garcke and Novick-Cohen as the singular limit of a Cahn-Hilliard equation with degenerated mobility. These conditions are the concurrency of the triple junction, angle conditions between the hypersurfaces, continuity of the chemical potentials and a flux-balance. For this system we show stability of its energy minimizers, i.e., standard double bubbles.The main argument relies on a Lojasiewicz-Simon gradient inequality. The proof of it differs from others works due to the fully non-linear boundary conditions and problems with the (non-local) tangential part.

math.AP↗