https://neckpinch.com/authors/helmut-abels/New arXiv papers by Helmut Abels, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2410.08596arxiv:2410.08596Fri, 11 Oct 2024 00:00:00 GMTHelmut Abels, Christoph Hurm, Maximilian Moser — We prove convergence of the nonlocal Allen-Cahn equation to mean curvature flow in the sharp interface limit, in the situation when the parameter corresponding to the kernel goes to zero fast enough with respect to the diffuse interface thickness. The analysis is done in the case of a W^(1,1)-kernel, under periodic boundary conditions and in both two and three space dimensions. We use the approximate solution and spectral estimate from the local case, and combine the latter with an L²-estimate for the difference of the nonlocal operator and the negative Laplacian from Abels, Hurm arXiv:2307.02264. To this end, we prove a nonlocal Ehrling-type inequality to show uniform H³-estimates for the nonlocal solutions.Mean curvature flowBoundaries & non-compactEigenvalues & spectrumPhase fields & multiphase flows https://arxiv.org/abs/2205.02493arxiv:2205.02493Thu, 05 May 2022 00:00:00 GMTHelmut Abels, Felicitas Bürger, Harald Garcke — We consider a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. Several properties of solutions are analyzed. Emphasis is placed on to what extent the surface in our setting qualitatively evolves similar as for the usual mean curvature flow. To this end, we show that the surface area is strictly decreasing but give an example of a surface that exists for infinite times nevertheless. Moreover, mean convexity is conserved whereas convexity is not. Finally, we construct an embedded hypersurface that develops a self-intersection in the course of time. Additionally, a formal explanation of how our equations can be interpreted as a gradient flow is included.Mean curvature flowSingularities https://arxiv.org/abs/2204.07626arxiv:2204.07626Fri, 15 Apr 2022 00:00:00 GMTHelmut Abels, Felicitas Bürger, Harald Garcke — We prove a short time existence result for a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss a mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. The proof is based on a splitting ansatz, solving both equations separately using linearization and a contraction argument. Our result is formulated for the case of immersed hypersurfaces and yields a uniform lower bound on the existence time that allows for small changes in the initial value of the height function.Mean curvature flowHarmonic maps & DeTurck trick