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Publications Tim Laux

Preprints

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39
The Verigin problem with phase transition as a Wasserstein flow, with Anna Kubin and Alice Marveggio, submitted, 26 pp.
https://arxiv.org/abs/2601.20001
38
Obstacle mean curvature flow: Efficient approximation and convergence analysis, with Fabius Krämer, submitted, 17 pp.
https://arxiv.org/abs/2512.16668
37
Equivalence of weak solution concepts for mean curvature flow, with Anton Ullrich, submitted, 13 pp.
https://arxiv.org/abs/2510.16478
36
Generic mean curvature flow with obstacles, with Keisuke Takasao, submitted, 25 pp.
https://arxiv.org/abs/2507.06150
35
Change of bifurcation type in 2 D free boundary model of a moving cell with nonlinear diffusion, with Leonid Berlyand and Oleksii Krupchytskyi, in revision, 32 pp.
https://arxiv.org/abs/2506.03138
34
An efficient volume-preserving MBO scheme for data clustering and classification,
with Fabius Krämer, submitted, 56 pp.
https://arxiv.org/abs/2412.17694
33
Median filter method for mean curvature flow using a random Jacobi algorithm,
with Anton Ullrich,
submitted, 56 pp.
https://arxiv.org/abs/2410.07776
32
A weak-strong uniqueness principle for the Mullins-Sekerka equation,
with Julian Fischer, Sebastian Hensel, and Theresa M. Simon,
preprint, 48 pp.
https://arxiv.org/abs/2404.02682
31
Local minimizers of the interface length functional based on a concept of local paired calibrations,
with Julian Fischer, Sebastian Hensel, and Theresa M. Simon,
submitted, 35 pp.
https://arxiv.org/abs/2212.11840

Publications in Peer-Reviewed Journals

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30
A uniqueness and stability principle for surface diffusion,
with Milan Krömer,
to appear in Interfaces Free Bound., 26 pp.
https://arxiv.org/abs/2212.12487
29
Large data limit of the MBO scheme for data clustering: Γ-convergence of the thresholding energies,
with Jona Lelmi,
Appl. Comp. Harm. Anal., Vol. 79, 2025, 101800, 34 pp.
https://doi.org/10.1016/j.acha.2025.101800
28
A mean curvature flow arising in adversarial training
with Leon Bungert and Kerrek Stinson, J. Math. Pures Appl., Vol. 192, 2024, 103625, 32 pp.
https://doi.org/10.1016/j.matpur.2024.103625
27
Quantitative convergence of the nonlocal Allen-Cahn equation to volume-preserving mean curvature flow,
with Milan Krömer, Math. Ann. 391, pp. 4455–4472, 2025.
https://doi.org/10.1007/s00208-024-03034-0
26
Generic level sets in mean curvature flow are BV solutions,
with Anton Ullrich, J. Geom. Anal. 34, 375, 16 pp., 2025
https://doi.org/10.1007/s12220-024-01819-y
25
Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow,
with Kerrek Stinson and Clemens Ullrich,
European J. Appl. Math. 36(1): 82–142, 2025.
  https://doi.org/10.1017/S0956792524000226
24
A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness,
with Sebastian Hensel, J. Differential Geom. 130(1):209–268, 2025.
https://doi.org/10.4310/jdg/1747065796
23
The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions,
with Julian Fischer, Sebastian Hensel, and Theresa M. Simon,
J. Eur. Math. Soc. (JEMS) online first, 104 pp.
https://doi.org/10.4171/JEMS/1577
22
Large data limit of the MBO scheme for data clustering: Convergence of the dynamics,
with Jona Lelmi,
J. Mach. Learn. Res. (JMLR) 24(344):1-49, 2023
https://jmlr.org/papers/v24/22-1089.html
21
A phase-field version of the Faber-Krahn theorem,
with Paul Hüttl and Patrik Knopf,
Interfaces Free Bound. 26, no. 4, pp. 587–623, 2024.
https://doi.org/10.4171/IFB/519
20
Sharp interface limit of the Cahn-Hilliard reaction model for lithium-ion batteries,
with Kerrek Stinson,
Math. Models Methods Appl. Sci., 33:12, 2557–2585, 2023.
https://doi.org/10.1142/S0218202523500550
19
Phase-field methods for spectral shape and topology optimization,
with Harald Garcke, Paul Hüttl, Christian Kahle, and Patrik Knopf,
ESAIM: Control Optim. Calc. Var. 29:10, 57 pp., 2023.
https://doi.org/10.1051/cocv/2022090
18
Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets,
with Jakob Fuchs,
Adv. Calc. Var. 17(2):421--465, 2024.
https://doi.org/10.1515/acv-2022-0020
17
Weak-strong uniqueness for volume-preserving mean curvature flow,
Rev. Mat. Iberoam. 40(1):93–110, 2024.
https://ems.press/journals/rmi/articles/8082294
16
The Hele-Shaw flow as the sharp interface limit of the Cahn-Hilliard equation with disparate mobilities,
with Milan Krömer,
Comm. Partial Differential Equations, 47(12):2444–2486, 2022.
https://www.tandfonline.com/doi/full/10.1080/03605302.2022.2129384
15
BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness,
with Sebastian Hensel,
Indiana Univ. Math. J. (online first), 24 pp.
https://arxiv.org/abs/2112.11150
14
Weak-strong uniqueness for the mean curvature flow of double bubbles,
with Sebastian Hensel,
Interfaces Free Bound. 25, no. 1, pp. 37–107, 2023.
https://doi.org/10.4171/ifb/484
13
De Giorgi's inequality for the thresholding scheme with arbitrary mobilities and surface tensions,
with Jona Lelmi,
Calc. Var. Partial Differential Equations, 61(1):35, 42 pp., 2022.
https://link.springer.com/article/10.1007/s00526-021-02146-8
12
Nematic-isotropic phase transition in liquid crystals: A variational derivation of effective geometric motions,
with Yuning Liu,
Arch. Ration. Mech. Anal. 241(3):1785–1814, 2021.
https://link.springer.com/article/10.1007/s00205-021-01681-0
11
Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies,
with Julian Fischer and Theresa M. Simon,
SIAM J. Math. Anal., 52(6):6222–6233, 2020.
https://epubs.siam.org/doi/abs/10.1137/20M1322182?af=R&mobileUi=0
10
Mullins-Sekerka as the Wasserstein flow of the perimeter,
with Antonin Chambolle,
Proc. Amer. Math. Soc., 149(7):2943–2956, 2021.
https://www.ams.org/journals/proc/2021-149-07/S0002-9939-2021-15401-4/
9
Implicit time discretization for the mean curvature flow of mean convex sets,
with Guido De Philippis,
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 21:911–930, 2020.
https://journals.sns.it/index.php/annaliscienze/article/view/924
8
Well-posedness for degenerate elliptic PDE arising in optimal learning strategies,
with J. Miguel Villas-Boas,
Interfaces Free Bound., 22(1):119–129, 2020.
https://www.ems-ph.org/journals/show_abstract.php?issn=1463-9963&vol=22&iss=1&rank=5
7
Brakke's inequality for the thresholding scheme,
with Felix Otto,
Calc. Var. Partial Differential Equations, 59(1):39, 26 pp., 2020.
https://link.springer.com/content/pdf/10.1007/s00526-020-1696-8.pdf
6
Analysis of diffusion generated motion for mean curvature flow in codimension two: A gradient-flow approach,
with Aaron Yip,
Arch. Ration. Mech. Anal., 232(2):1113–1163, 2019.
http://link.springer.com/article/10.1007/s00205-018-01340-x
5
Convergence of the Allen-Cahn equation to multiphase mean curvature flow,
with Theresa M. Simon,
Comm. Pure Appl. Math., 71(8):1597–1647, 2018.
http://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.21747
4
Gradient-flow techniques for the analysis of numerical schemes for multi-phase mean-curvature flow,
Geometric Flows, 3(1):76–89, 2018.
http://www.degruyter.com/view/j/geofl.2018.3.issue-1/geofl-2018-0006/geofl-2018-0006.xml
3
The elastic flow of curves on the sphere,
with Anna Dall'Acqua, Chun-Chi Lin, Paola Pozzi, and Adrian Spener,
Geometric Flows, 3(1):1–13, 2018.
http://www.degruyter.com/view/j/geofl.2018.3.issue-1/geofl-2018-0001/geofl-2018-0001.xml
2
Convergence of thresholding schemes incorporating bulk effects,
with Drew Swartz,
Interfaces Free Bound., 19(2):273–304, 2017.
http://www.ems-ph.org/journals/show_abstract.php?issn=1463-9963&vol=19&iss=2&rank=5
1
Convergence of the thresholding scheme for multi-phase mean-curvature flow,
with Felix Otto,
Calc. Var. Partial Differential Equations, 55(5):129, 74 pp., 2016.
http://link.springer.com/article/10.1007%2Fs00526-016-1053-0

Reports and Proceedings

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P9
The gradient-flow structure of the Verigin problem and its natural time discretization, with Anna Kubin and Alice Marveggio. Oberwolfach Report 22 (2025), no. 2, pp. 1208–1209.
https://doi.org/10.4171/OWR/2025/23
P8
The large-data limit of the MBO scheme for data clustering,
with Jona Lelmi,
Oberwolfach Report 20 (2023), no. 4, pp. 3190–3193.
https://doi.org/10.4171/OWR/2023/57
P7
A new varifold solution concept for mean curvature flow,
with Sebastian Hensel,
Oberwolfach Report 20 (2023), no. 3, pp. 1831–1834.
https://doi.org/10.4171/OWR/2023/32
P6
Large data limit of the MBO scheme for data clustering,
with Jona Lelmi,
Proc. Appl. Math. Mech. 22.1, e202200308, 2023.
https://doi.org/10.1002/pamm.202200308
P5
A gradient-flow approach for the convergence of the anisotropic Allen-Cahn equation.

http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/2172-03.pdf

, Geometric Aspects of Solutions to Partial Differential Equations, pp. 32-42, Research Institute for Mathematical Sciences, Kyoto University, Japan, 2020.
P4
The thresholding scheme for mean curvature flow and De Giorgi's ideas for minimizing movements,
with Felix Otto,

https://projecteuclid.org/euclid.aspm/1609261543

, The Role of Metrics in the Theory of Partial Differential Equations, pp. 63-93, Mathematical Society of Japan, 2020.
P3
Thresholding for mean curvature flow in codimension two,
with Aaron Yip,
Oberwolfach Report, OWR 3/2019, pp. 147-150.
https://ems.press/journals/owr/articles/16765
P2
Brakke – de Giorgi – Osher,
with Felix Otto,
Oberwolfach Report, OWR 54/2017, pp. 3219-3222.
https://ems.press/journals/owr/articles/15959
P1
Kornwachstum in Polykristallen: Algorithmen für den mittleren Krümmungsfluss,
with Felix Otto,

http://www.mpg.de/10893983/mpi_mis_jb_2017?c=11356432&force_lang=de

, MPI for Mathematics in the Sciences, 2017.

Lecture Notes

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L1.
Distributional solutions to mean curvature flow, 38 pages
https://arxiv.org/abs/2108.08347

Outreach

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O1.
Sauer, J., 2022. Episode 8 - Tim Laux. Literatur-Rundschau.
https://doi.org/10.22032/dbt.59076
O2.
Die Mathematik des Kristallwachstums – ein Forschungsauftrag für Schüler*innen,
with Fabian Weidt and Stefan Hartmann
https://www.faszination.uni-bonn.de/de/modelle/kristallwachstum
O3.
Optimale Formen und Muster - Ist die Natur eine Mathematikerin?
Vorlesung im Rahmen der Schüler*innenwoche am Hausdorffzentrum für Mathematik der Universität Bonn
https://www.uni-regensburg.de/assets/mathematics/mathematics-laux/Skript_mit_Aufgaben.pdf

Theses

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T3.
Convergence of phase-field models and thresholding schemes via the gradient flow structure of multi-phase mean-curvature flow,
PhD Thesis, Max Planck Institute for Mathematics in the Sciences and University of Leipzig, 2017.
T2.
Dynamics of magnetic phase transitions,
Master's Thesis, RWTH Aachen University, 2013.
T1.
Maximum principles in differential inequalities and monotonicity of solutions,
Bachelor's Thesis, RWTH Aachen University, 2011.

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