--- OA_place: repository OA_type: green _id: '10011' abstract: - lang: eng text: 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. acknowledgement: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 948819), and from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813. The content of this paper was developed and parts of it were written during a visit of the first author to the Hausdorff Center of Mathematics (HCM), University of Bonn. The hospitality and the support of HCM are gratefully acknowledged. article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Sebastian full_name: Hensel, Sebastian id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87 last_name: Hensel orcid: 0000-0001-7252-8072 - first_name: Tim full_name: Laux, Tim last_name: Laux citation: ama: 'Hensel S, Laux T. A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness. Journal of Differential Geometry. 2025;130:209-268. doi:10.4310/jdg/1747065796' apa: 'Hensel, S., & Laux, T. (2025). A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness. Journal of Differential Geometry. International Press. https://doi.org/10.4310/jdg/1747065796' chicago: 'Hensel, Sebastian, and Tim Laux. “A New Varifold Solution Concept for Mean Curvature Flow: Convergence of  the Allen-Cahn Equation and Weak-Strong Uniqueness.” Journal of Differential Geometry. International Press, 2025. https://doi.org/10.4310/jdg/1747065796.' ieee: 'S. Hensel and T. Laux, “A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness,” Journal of Differential Geometry, vol. 130. International Press, pp. 209–268, 2025.' ista: 'Hensel S, Laux T. 2025. A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness. Journal of Differential Geometry. 130, 209–268.' mla: 'Hensel, Sebastian, and Tim Laux. “A New Varifold Solution Concept for Mean Curvature Flow: Convergence of  the Allen-Cahn Equation and Weak-Strong Uniqueness.” Journal of Differential Geometry, vol. 130, International Press, 2025, pp. 209–68, doi:10.4310/jdg/1747065796.' short: S. Hensel, T. Laux, Journal of Differential Geometry 130 (2025) 209–268. corr_author: '1' date_created: 2021-09-13T12:17:10Z date_published: 2025-05-01T00:00:00Z date_updated: 2025-05-28T09:27:05Z day: '01' department: - _id: JuFi doi: 10.4310/jdg/1747065796 ec_funded: 1 external_id: arxiv: - '2109.04233' intvolume: ' 130' keyword: - Mean curvature flow - gradient flows - varifolds - weak solutions - weak-strong uniqueness - calibrated geometry - gradient-flow calibrations language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/2109.04233 month: '05' oa: 1 oa_version: Preprint page: 209-268 project: - _id: 0aa76401-070f-11eb-9043-b5bb049fa26d call_identifier: H2020 grant_number: '948819' name: Bridging Scales in Random Materials publication: Journal of Differential Geometry publication_identifier: eissn: - 1945-743X issn: - 0022-040X publication_status: published publisher: International Press quality_controlled: '1' scopus_import: '1' status: public title: 'A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness' type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 130 year: '2025' ... --- OA_place: publisher _id: '20563' abstract: - lang: eng text: "The theory of optimal transport provides an elegant and powerful description of many evolution\r\nequations as gradient flows. The primary objective of this thesis is to adapt and extend the\r\ntheory to deal with important equations that are not covered by the classical framework,\r\nspecifically boundary value problems and kinetic equations. Additionally, we establish new\r\nresults in periodic homogenization for discrete dynamical optimal transport and in quantization\r\nof measures.\r\nSection 1.1 serves as an invitation to the classical theory of optimal transport, including the\r\nmain definitions and a selection of well-established theorems. Sections 1.2-1.5 introduce the\r\nmain results of this thesis, outline the motivations, and review the current state of the art.\r\nIn Chapter 2, we consider the Fokker–Planck equation on a bounded set with positive Dirichlet\r\nboundary conditions. We construct a time-discrete scheme involving a modification of the\r\nWasserstein distance and, under weak assumptions, prove its convergence to a solution of this\r\nboundary value problem. In dimension 1, we show that this solution is a gradient flow in a\r\nsuitable space of measures.\r\nChapter 3 presents joint work with Giovanni Brigati and Jan Maas. We introduce a new theory\r\nof optimal transport to describe and study particle systems at the mesoscopic scale. We prove\r\nadapted versions of some fundamental theorems, including the Benamou–Brenier formula and\r\nthe identification of absolutely continuous curves of measures.\r\nChapter 4 presents joint work with Lorenzo Portinale. We prove convergence of dynamical\r\ntransportation functionals on periodic graphs in the large-scale limit when the cost functional\r\nis asymptotically linear. Additionally, we show that discrete 1-Wasserstein distances converge\r\nto 1-Wasserstein distances constructed from crystalline norms on R\r\nd\r\n.\r\nChapter 5 concerns optimal empirical quantization: the problem of approximating a measure\r\nby the sum of n equally weighted Dirac deltas, so as to minimize the error in the p-Wasserstein\r\ndistance. Our main result is an analog of Zador’s theorem, providing asymptotic bounds for\r\nthe minimal error as n tends to infinity.\r\n" acknowledgement: "The research contained in this thesis has received funding from the Austrian Science\r\nFund (FWF) project 10.55776/F65." alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Filippo full_name: Quattrocchi, Filippo id: 3ebd6ba8-edfb-11eb-afb5-91a9745ba308 last_name: Quattrocchi orcid: 0009-0000-9773-1931 citation: ama: Quattrocchi F. Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures. 2025. doi:10.15479/AT-ISTA-20563 apa: Quattrocchi, F. (2025). Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures. Institute of Science and Technology Austria. https://doi.org/10.15479/AT-ISTA-20563 chicago: Quattrocchi, Filippo. “Optimal Transport Methods for Kinetic Equations, Boundary Value Problems, and Discretization of Measures.” Institute of Science and Technology Austria, 2025. https://doi.org/10.15479/AT-ISTA-20563. ieee: F. Quattrocchi, “Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures,” Institute of Science and Technology Austria, 2025. ista: Quattrocchi F. 2025. Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures. Institute of Science and Technology Austria. mla: Quattrocchi, Filippo. Optimal Transport Methods for Kinetic Equations, Boundary Value Problems, and Discretization of Measures. Institute of Science and Technology Austria, 2025, doi:10.15479/AT-ISTA-20563. short: F. Quattrocchi, Optimal Transport Methods for Kinetic Equations, Boundary Value Problems, and Discretization of Measures, Institute of Science and Technology Austria, 2025. corr_author: '1' date_created: 2025-10-28T13:10:49Z date_published: 2025-11-03T00:00:00Z date_updated: 2026-04-07T12:39:35Z day: '03' ddc: - '515' - '519' degree_awarded: PhD department: - _id: GradSch - _id: JaMa doi: 10.15479/AT-ISTA-20563 file: - access_level: open_access checksum: 6f55275bdf99992be3a6457d949dd664 content_type: application/pdf creator: fquattro date_created: 2025-11-17T21:04:15Z date_updated: 2026-01-01T23:30:03Z embargo: 2026-01-01 file_id: '20653' file_name: 2025_quattrocchi_filippo_thesis.pdf file_size: 4326411 relation: main_file - access_level: closed checksum: 707e580f5d993a214c0dba456b75837b content_type: application/zip creator: fquattro date_created: 2025-11-17T21:05:43Z date_updated: 2026-01-01T23:30:03Z embargo_to: open_access file_id: '20654' file_name: 2025_quattrocchi_thesis.zip file_size: 11726509 relation: source_file file_date_updated: 2026-01-01T23:30:03Z has_accepted_license: '1' keyword: - optimal transport - kinetic equations - boundary value problems - quantization - gradient flows - homogenization language: - iso: eng month: '11' oa: 1 oa_version: Published Version page: '240' project: - _id: 260482E2-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: F06504 name: Taming Complexity in Partial Differential Systems publication_identifier: issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria related_material: record: - id: '18706' relation: part_of_dissertation status: public - id: '20569' relation: part_of_dissertation status: public - id: '20571' relation: part_of_dissertation status: public - id: '20570' relation: part_of_dissertation status: public status: public supervisor: - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 title: Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: dissertation user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd year: '2025' ... --- OA_place: repository OA_type: green _id: '20571' abstract: - lang: eng text: "We prove the convergence of a modified Jordan--Kinderlehrer--Otto scheme to a solution to the Fokker--Planck equation in $\\Omega \\Subset \\mathbb{R}^d$ with general, positive and temporally constant, Dirichlet boundary conditions. We work under mild assumptions on the domain, the drift, and the initial datum. \ In the special case where $\\Omega$ is an interval in $\\mathbb{R}^1$, we prove that such a solution is a gradient flow -- curve of maximal slope -- within a suitable space of measures, endowed with a modified Wasserstein distance.\r\nOur discrete scheme and modified distance draw inspiration from contributions by A. Figalli and N. Gigli [J. Math. Pures Appl. 94, (2010), pp. 107--130], and J. Morales [J. Math. Pures Appl. 112, (2018), pp. 41--88] on an optimal-transport approach to evolution equations with Dirichlet boundary conditions. Similarly to these works, we allow the mass to flow from/to the boundary $\\partial \\Omega$ throughout the evolution. However, our leading idea is to also keep track of the mass at the boundary by working with measures defined on the whole closure $\\overline \\Omega$. The driving functional is a modification of the classical relative entropy that also makes use of the information at the boundary. As an intermediate result, when $\\Omega$ is an interval in $\\mathbb{R}^1$, we find a formula for the descending slope of this geodesically nonconvex functional. " acknowledgement: "The author would like to thank Jan Maas for suggesting this project and for many helpful\r\ncomments, Antonio Agresti, Lorenzo Dello Schiavo and Julian Fischer for several fruitful discussions, and Oliver Tse for pointing out the reference [15]. He also gratefully acknowledges support from the Austrian Science Fund (FWF) project 10.55776/F65.\r\n" article_number: '2403.07803' article_processing_charge: No arxiv: 1 author: - first_name: Filippo full_name: Quattrocchi, Filippo id: 3ebd6ba8-edfb-11eb-afb5-91a9745ba308 last_name: Quattrocchi orcid: 0009-0000-9773-1931 citation: ama: Quattrocchi F. Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. arXiv. doi:10.48550/arXiv.2403.07803 apa: Quattrocchi, F. (n.d.). Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. arXiv. https://doi.org/10.48550/arXiv.2403.07803 chicago: Quattrocchi, Filippo. “Variational Structures for the Fokker-Planck Equation with General Dirichlet Boundary Conditions.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2403.07803. ieee: F. Quattrocchi, “Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions,” arXiv. . ista: Quattrocchi F. Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. arXiv, 2403.07803. mla: Quattrocchi, Filippo. “Variational Structures for the Fokker-Planck Equation with General Dirichlet Boundary Conditions.” ArXiv, 2403.07803, doi:10.48550/arXiv.2403.07803. short: F. Quattrocchi, ArXiv (n.d.). corr_author: '1' date_created: 2025-10-28T13:12:56Z date_published: 2024-04-09T00:00:00Z date_updated: 2026-06-04T22:30:22Z day: '09' department: - _id: GradSch - _id: JaMa doi: 10.48550/arXiv.2403.07803 external_id: arxiv: - '2403.07803' keyword: - gradient flows - Jordan–Kinderlehrer–Otto scheme - curves of maximal slope - optimal transport - Dirichlet boundary conditions - Fokker–Planck equation language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2403.07803 month: '04' oa: 1 oa_version: Preprint project: - _id: 260482E2-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: F06504 name: Taming Complexity in Partial Differential Systems publication: arXiv publication_status: draft related_material: record: - id: '20865' relation: later_version status: public - id: '20563' relation: dissertation_contains status: public status: public title: Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2024' ...