https://neckpinch.com/authors/julian-fischer/New arXiv papers by Julian Fischer, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2404.02884arxiv:2404.02884Wed, 03 Apr 2024 00:00:00 GMTJulian Fischer, Sebastian Hensel, Alice Marveggio, Maximilian Moser — We prove a weak-strong uniqueness principle for varifold-BV solutions to planar multiphase mean curvature flow beyond a circular topology change: Assuming that there exists a classical solution with an interface that becomes increasingly circular and shrinks to a point, any varifold-BV solution with the same initial interface must coincide with it, and any varifold-BV solution with similar initial data must undergo the same type of topology change. Our result illustrates the robustness of the relative energy method for establishing weak-strong uniqueness principles for interface evolution equations, showing that it may also be applied beyond certain topological changes.Mean curvature flowWeak & singular flowsStability & uniquenessPhase fields & multiphase flows https://arxiv.org/abs/2203.17143arxiv:2203.17143Thu, 31 Mar 2022 00:00:00 GMTJulian Fischer, Alice Marveggio — Phase-field models such as the Allen-Cahn equation may give rise to the formation and evolution of geometric shapes, a phenomenon that may be analyzed rigorously in suitable scaling regimes. In its sharp-interface limit, the vectorial Allen-Cahn equation with a potential with N≥ 3 distinct minima has been conjectured to describe the evolution of branched interfaces by multiphase mean curvature flow. In the present work, we give a rigorous proof for this statement in two and three ambient dimensions and for a suitable class of potentials: As long as a strong solution to multiphase mean curvature flow exists, solutions to the vectorial Allen-Cahn equation with well-prepared initial data converge towards multiphase mean curvature flow in the limit of vanishing interface width parameter ε↘ 0. We even establish the rate of convergence O(ε^(1/2)). Our approach is based on the gradient flow structure of the Allen-Cahn equation and its limiting motion: Building on the recent concept of "gradient flow calibrations" for multiphase mean curvature flow, we introduce a notion of relative entropy for the vectorial Allen-Cahn equation with multi-well potential. This enables us to overcome the limitations of other approaches, e.g. avoiding the need for a stability analysis of the Allen-Cahn operator or additional convergence hypotheses for the energy at positive times.Mean curvature flowStability & uniquenessPhase fields & multiphase flows