https://neckpinch.com/authors/tim-laux/New arXiv papers by Tim Laux, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.21545arxiv:2610.21545Thu, 15 Oct 2026 00:00:00 GMTFabius Krämer, Tim Laux, Keisuke Takasao — We prove the generic existence of mean curvature flows in spaces with rough weights. Our weak solutions are both unit-density Brakke flows and distributional solutions. Such a generic statement is optimal in the sense that we consider weights that are too rough to have well-defined traces on every hypersurface.Mean curvature flowWeak & singular flows https://arxiv.org/abs/2608.26842arxiv:2608.26842Thu, 27 Aug 2026 00:00:00 GMTTim Laux, Keisuke Takasao — Since Ilmanen's pioneering work [J. Differential Geom. 38, 417-461, (1993)] it has been a major open problem to understand the sharp-interface limit of systems of coupled Allen-Cahn equations. We prove - for the first time - a global, unconditional convergence result for such a coupled system and show that the limit is a multiphase mean curvature flow in the sense of Brakke.Mean curvature flowWeak & singular flowsPhase fields & multiphase flows https://arxiv.org/abs/2607.03930arxiv:2607.03930Sat, 04 Jul 2026 00:00:00 GMTTim Laux, Andrea Poiatti — We propose an alternative existence proof of global weak solutions to mean curvature flow and volume preserving mean curvature flow. We prove for the first time for a minimizing movements scheme the unconditional convergence towards a varifold solution, here a De Giorgi solution. The argument is purely variational and does not rely on comparison principles. The key novelty is an alternative proxy for the completely degenerate L² distance that is more robust than the one of Almgren-Taylor-Wang and Luckhaus-Sturzenhecker.Mean curvature flowVolume-preserving & constrained flowsWeak & singular flows https://arxiv.org/abs/2510.16478arxiv:2510.16478Sat, 18 Oct 2025 00:00:00 GMTTim Laux, Anton Ullrich — We provide a connection between weak solution concepts of mean curvature flow. On the one side we have the viscosity solution which is based on the comparison principle. On the other, variational solutions, which are combined Brakke flows and distributional solutions. We prove that if one has a foliation by variational solutions, then the resulting function is the unique viscosity solution. This answers an open question suggested by the work of Evans and Spruck [J. Geom. Anal., 5(1), 1995] and the authors [J. Geom. Anal., 34(12), 2024]. These results show that almost every level set of the viscosity solution is a variational solution. Thus, we establish the equivalence of these solution concepts. Moreover, we show the generic uniqueness of variational solutions.Mean curvature flowWeak & singular flowsStability & uniqueness https://arxiv.org/abs/2507.06150arxiv:2507.06150Tue, 08 Jul 2025 00:00:00 GMTTim Laux, Keisuke Takasao — We study the obstacle problem associated to mean curvature flow. We add to the geometric vanishing-viscosity approximation of Evans and Spruck a singular perturbation that penalizes the violation of the constraint, and pass to the limit. The resulting level set formulation has unique solutions - up to fattening. Extending the work of Evans and Spruck and a work by Ullrich and one of the authors, we show that generic level sets of this flow are distributional solutions of the obstacle problem.Mean curvature flowWeak & singular flows https://arxiv.org/abs/2410.07776arxiv:2410.07776Thu, 10 Oct 2024 00:00:00 GMTAnton Ullrich, Tim Laux — We present an efficient scheme for level set mean curvature flow using a domain discretization and median filters. For this scheme, we show convergence in L^∞-norm under mild assumptions on the number of points in the discretization. In addition, we strengthen the weak convergence result for the MBO thresholding scheme applied to data clustering of Lelmi and one of the authors. This is done through a strong convergence of the discretized heat flow in the optimal regime. Different boundary conditions are also discussed.Mean curvature flowBoundaries & non-compactPhase fields & multiphase flows https://arxiv.org/abs/2404.14402arxiv:2404.14402Mon, 22 Apr 2024 00:00:00 GMTLeon Bungert, Tim Laux, Kerrek Stinson — We connect adversarial training for binary classification to a geometric evolution equation for the decision boundary. Relying on a perspective that recasts adversarial training as a regularization problem, we introduce a modified training scheme that constitutes a minimizing movements scheme for a nonlocal perimeter functional. We prove that the scheme is monotone and consistent as the adversarial budget vanishes and the perimeter localizes, and as a consequence we rigorously show that the scheme approximates a weighted mean curvature flow. This highlights that the efficacy of adversarial training may be due to locally minimizing the length of the decision boundary. In our analysis, we introduce a variety of tools for working with the subdifferential of a supremal-type nonlocal total variation and its regularity properties.Mean curvature flowAnisotropic & nonlocal flowsWeak & singular flows https://arxiv.org/abs/2309.12409arxiv:2309.12409Thu, 21 Sep 2023 00:00:00 GMTMilan Kroemer, Tim Laux — We prove a quantitative convergence result of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow. The proof uses gradient flow calibrations and the relative entropy method, which has been used in the recent literature to prove weak-strong uniqueness results for mean curvature flow and convergence of the Allen–Cahn equation. A crucial difference in this work is a new notion of gradient flow calibrations. We add a tangential component to the velocity field in order to prove the Gronwall estimate for the relative energy. This allows us to derive the optimal convergence rate without having to show the closeness of the Lagrange-multipliers.Mean curvature flowVolume-preserving & constrained flowsWeak & singular flowsStability & uniquenessPhase fields & multiphase flows https://arxiv.org/abs/2301.01097arxiv:2301.01097Tue, 03 Jan 2023 00:00:00 GMTAnton Ullrich, Tim Laux — We show that a generic levelset of the viscosity solution to mean curvature flow is a distributional solution in the framework of sets of finite perimeter by Luckhaus and Sturzenhecker, which in addition saturates the optimal energy dissipation rate. This extends the fundamental work of Evans and Spruck (J. Geom. Anal. 1995), which draws a similar connection between the viscosity solution and Brakke flows.Mean curvature flowWeak & singular flows https://arxiv.org/abs/2212.11939arxiv:2212.11939Thu, 22 Dec 2022 00:00:00 GMTTim Laux, Kerrek Stinson, Clemens Ullrich — The purpose of this paper is to derive anisotropic mean curvature flow as the limit of the anisotropic Allen-Cahn equation. We rely on distributional solution concepts for both the diffuse and sharp interface models, and prove convergence using relative entropy methods, which have recently proven to be a powerful tool in interface evolution problems. With the same relative entropy, we prove a weak-strong uniqueness result, which relies on the construction of gradient flow calibrations for our anisotropic energy functionals.Mean curvature flowAnisotropic & nonlocal flowsWeak & singular flowsStability & uniquenessPhase fields & multiphase flows https://arxiv.org/abs/2205.13040arxiv:2205.13040Wed, 25 May 2022 00:00:00 GMTTim Laux — In this note, we derive a stability and weak-strong uniqueness principle for volume-preserving mean curvature flow. The proof is based on a new notion of volume-preserving gradient flow calibrations, which is a natural extension of the concept in the case without volume preservation recently introduced by Fischer et al. [arXiv:2003.05478]. The first main result shows that any strong solution with certain regularity is calibrated. The second main result consists of a stability estimate in terms of a relative entropy, which is valid in the class of distributional solutions to volume-preserving mean curvature flow.Mean curvature flowVolume-preserving & constrained flowsWeak & singular flowsStability & uniqueness https://arxiv.org/abs/2201.00413arxiv:2201.00413Sun, 02 Jan 2022 00:00:00 GMTJakob Fuchs, Tim Laux — In this work, we analyze Merriman, Bence and Osher's thresholding scheme, a time discretization for mean curvature flow. We restrict to the two-phase setting and mean convex initial conditions. In the sense of the minimizing movements interpretation of Esedoglu and Otto we show the time-integrated energy of the approximation to converge to the time-integrated energy of the limit. As a corollary, the conditional convergence results of Otto and one of the authors become unconditional in the two-phase mean convex case. Our results are general enough to handle the extension of the scheme to anisotropic flows for which a non-negative kernel can be chosen.Mean curvature flowAnisotropic & nonlocal flowsWeak & singular flowsPhase fields & multiphase flows https://arxiv.org/abs/2112.11150arxiv:2112.11150Tue, 21 Dec 2021 00:00:00 GMTSebastian Hensel, Tim Laux — We study weak solutions to mean curvature flow satisfying Young's angle condition for general contact angles α∈ (0,π). First, we construct BV solutions using the Allen-Cahn approximation with boundary contact energy as proposed by Owen and Sternberg. Second, we prove the weak-strong uniqueness and stability for this solution concept. The main ingredient for both results is a relative energy, which can also be interpreted as a tilt excess.Mean curvature flowWeak & singular flowsStability & uniquenessFree boundary & capillaryPhase fields & multiphase flows https://arxiv.org/abs/2109.04233arxiv:2109.04233Thu, 09 Sep 2021 00:00:00 GMTSebastian Hensel, Tim Laux — We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478] - and hence any classical solution to mean curvature flow - is unique in the class of our new varifold solutions. This is in sharp contrast to the case of Brakke flows, which a priori may disappear at any given time and are therefore fatally non-unique. Finally, we propose an extension of the solution concept to the multi-phase case which is at least guaranteed to satisfy a weak-strong uniqueness principle.Mean curvature flowWeak & singular flowsStability & uniquenessPhase fields & multiphase flows https://arxiv.org/abs/2108.08347arxiv:2108.08347Wed, 18 Aug 2021 00:00:00 GMTTim Laux — These lecture notes aim to present some of the ideas behind the recent (conditional) existence and (weak-strong) uniqueness theory for mean curvature flow. Focusing on the simplest case of the evolution of a single closed hypersurface allows for a self-contained and concise presentation, which is accessible for beginning graduate students with some background in PDEs and only requires basic measure theory.Mean curvature flowWeak & singular flowsStability & uniqueness https://arxiv.org/abs/2108.01733arxiv:2108.01733Tue, 03 Aug 2021 00:00:00 GMTSebastian Hensel, Tim Laux — We derive a weak-strong uniqueness principle for BV solutions to multiphase mean curvature flow of triple line clusters in three dimensions. Our proof is based on the explicit construction of a gradient-flow calibration in the sense of the recent work of Fischer et al. [arXiv:2003.05478v2] for any such cluster. This extends the two-dimensional construction to the three-dimensional case of surfaces meeting along triple junctions.Mean curvature flow3-manifoldsSurfacesWeak & singular flowsStability & uniquenessPhase fields & multiphase flows