[{"arxiv":1,"intvolume":"       130","department":[{"_id":"JuFi"}],"oa":1,"date_updated":"2025-05-28T09:27:05Z","project":[{"grant_number":"948819","_id":"0aa76401-070f-11eb-9043-b5bb049fa26d","name":"Bridging Scales in Random Materials","call_identifier":"H2020"}],"author":[{"first_name":"Sebastian","last_name":"Hensel","id":"4D23B7DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-7252-8072","full_name":"Hensel, Sebastian"},{"last_name":"Laux","first_name":"Tim","full_name":"Laux, Tim"}],"month":"05","date_published":"2025-05-01T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"journal_article","status":"public","article_type":"original","quality_controlled":"1","doi":"10.4310/jdg/1747065796","keyword":["Mean curvature flow","gradient flows","varifolds","weak solutions","weak-strong uniqueness","calibrated geometry","gradient-flow calibrations"],"citation":{"apa":"Hensel, S., &#38; Laux, T. (2025). A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness. <i>Journal of Differential Geometry</i>. International Press. <a href=\"https://doi.org/10.4310/jdg/1747065796\">https://doi.org/10.4310/jdg/1747065796</a>","short":"S. Hensel, T. Laux, Journal of Differential Geometry 130 (2025) 209–268.","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.","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,” <i>Journal of Differential Geometry</i>, vol. 130. International Press, pp. 209–268, 2025.","ama":"Hensel S, Laux T. A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness. <i>Journal of Differential Geometry</i>. 2025;130:209-268. doi:<a href=\"https://doi.org/10.4310/jdg/1747065796\">10.4310/jdg/1747065796</a>","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.” <i>Journal of Differential Geometry</i>. International Press, 2025. <a href=\"https://doi.org/10.4310/jdg/1747065796\">https://doi.org/10.4310/jdg/1747065796</a>.","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.” <i>Journal of Differential Geometry</i>, vol. 130, International Press, 2025, pp. 209–68, doi:<a href=\"https://doi.org/10.4310/jdg/1747065796\">10.4310/jdg/1747065796</a>."},"main_file_link":[{"url":"https://arxiv.org/abs/2109.04233","open_access":"1"}],"publisher":"International Press","external_id":{"arxiv":["2109.04233"]},"page":"209-268","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."}],"_id":"10011","oa_version":"Preprint","scopus_import":"1","date_created":"2021-09-13T12:17:10Z","language":[{"iso":"eng"}],"OA_type":"green","volume":130,"publication_identifier":{"eissn":["1945-743X"],"issn":["0022-040X"]},"OA_place":"repository","day":"01","year":"2025","publication":"Journal of Differential Geometry","article_processing_charge":"No","title":"A new varifold solution concept for mean curvature flow: Convergence of  the Allen-Cahn equation and weak-strong uniqueness","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.","corr_author":"1","publication_status":"published","ec_funded":1},{"publisher":"Institute of Science and Technology Austria","degree_awarded":"PhD","page":"240","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"}],"_id":"20563","oa_version":"Published Version","has_accepted_license":"1","doi":"10.15479/AT-ISTA-20563","keyword":["optimal transport","kinetic equations","boundary value problems","quantization","gradient flows","homogenization"],"license":"https://creativecommons.org/licenses/by/4.0/","citation":{"ista":"Quattrocchi F. 2025. Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures. Institute of Science and Technology Austria.","short":"F. Quattrocchi, Optimal Transport Methods for Kinetic Equations, Boundary Value Problems, and Discretization of Measures, Institute of Science and Technology Austria, 2025.","apa":"Quattrocchi, F. (2025). <i>Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT-ISTA-20563\">https://doi.org/10.15479/AT-ISTA-20563</a>","ama":"Quattrocchi F. Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures. 2025. doi:<a href=\"https://doi.org/10.15479/AT-ISTA-20563\">10.15479/AT-ISTA-20563</a>","chicago":"Quattrocchi, Filippo. “Optimal Transport Methods for Kinetic Equations, Boundary Value Problems, and Discretization of Measures.” Institute of Science and Technology Austria, 2025. <a href=\"https://doi.org/10.15479/AT-ISTA-20563\">https://doi.org/10.15479/AT-ISTA-20563</a>.","mla":"Quattrocchi, Filippo. <i>Optimal Transport Methods for Kinetic Equations, Boundary Value Problems, and Discretization of Measures</i>. Institute of Science and Technology Austria, 2025, doi:<a href=\"https://doi.org/10.15479/AT-ISTA-20563\">10.15479/AT-ISTA-20563</a>.","ieee":"F. Quattrocchi, “Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures,” Institute of Science and Technology Austria, 2025."},"author":[{"last_name":"Quattrocchi","first_name":"Filippo","full_name":"Quattrocchi, Filippo","orcid":"0009-0000-9773-1931","id":"3ebd6ba8-edfb-11eb-afb5-91a9745ba308"}],"date_published":"2025-11-03T00:00:00Z","month":"11","type":"dissertation","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","status":"public","department":[{"_id":"GradSch"},{"_id":"JaMa"}],"date_updated":"2026-04-07T12:39:35Z","oa":1,"project":[{"_id":"260482E2-B435-11E9-9278-68D0E5697425","name":"Taming Complexity in Partial Differential Systems","call_identifier":"FWF","grant_number":"F06504"}],"related_material":{"record":[{"id":"18706","relation":"part_of_dissertation","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"20569"},{"relation":"part_of_dissertation","status":"public","id":"20571"},{"id":"20570","relation":"part_of_dissertation","status":"public"}]},"alternative_title":["ISTA Thesis"],"article_processing_charge":"No","acknowledgement":"The research contained in this thesis has received funding from the Austrian Science\r\nFund (FWF) project 10.55776/F65.","title":"Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures","corr_author":"1","publication_status":"published","supervisor":[{"full_name":"Maas, Jan","orcid":"0000-0002-0845-1338","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","last_name":"Maas","first_name":"Jan"}],"publication_identifier":{"issn":["2663-337X"]},"file_date_updated":"2026-01-01T23:30:03Z","OA_place":"publisher","day":"03","year":"2025","ddc":["515","519"],"date_created":"2025-10-28T13:10:49Z","language":[{"iso":"eng"}],"tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"file":[{"file_name":"2025_quattrocchi_filippo_thesis.pdf","relation":"main_file","file_size":4326411,"checksum":"6f55275bdf99992be3a6457d949dd664","creator":"fquattro","embargo":"2026-01-01","file_id":"20653","content_type":"application/pdf","date_updated":"2026-01-01T23:30:03Z","date_created":"2025-11-17T21:04:15Z","access_level":"open_access"},{"file_id":"20654","embargo_to":"open_access","content_type":"application/zip","date_updated":"2026-01-01T23:30:03Z","access_level":"closed","date_created":"2025-11-17T21:05:43Z","file_name":"2025_quattrocchi_thesis.zip","file_size":11726509,"relation":"source_file","checksum":"707e580f5d993a214c0dba456b75837b","creator":"fquattro"}]},{"_id":"20571","oa_version":"Preprint","external_id":{"arxiv":["2403.07803"]},"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. "}],"citation":{"ista":"Quattrocchi F. Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. arXiv, 2403.07803.","short":"F. Quattrocchi, ArXiv (n.d.).","apa":"Quattrocchi, F. (n.d.). Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2403.07803\">https://doi.org/10.48550/arXiv.2403.07803</a>","mla":"Quattrocchi, Filippo. “Variational Structures for the Fokker-Planck Equation with General Dirichlet Boundary Conditions.” <i>ArXiv</i>, 2403.07803, doi:<a href=\"https://doi.org/10.48550/arXiv.2403.07803\">10.48550/arXiv.2403.07803</a>.","chicago":"Quattrocchi, Filippo. “Variational Structures for the Fokker-Planck Equation with General Dirichlet Boundary Conditions.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2403.07803\">https://doi.org/10.48550/arXiv.2403.07803</a>.","ama":"Quattrocchi F. Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2403.07803\">10.48550/arXiv.2403.07803</a>","ieee":"F. Quattrocchi, “Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions,” <i>arXiv</i>. ."},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2403.07803"}],"keyword":["gradient flows","Jordan–Kinderlehrer–Otto scheme","curves of maximal slope","optimal transport","Dirichlet boundary conditions","Fokker–Planck equation"],"doi":"10.48550/arXiv.2403.07803","status":"public","type":"preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"last_name":"Quattrocchi","first_name":"Filippo","full_name":"Quattrocchi, Filippo","id":"3ebd6ba8-edfb-11eb-afb5-91a9745ba308","orcid":"0009-0000-9773-1931"}],"date_published":"2024-04-09T00:00:00Z","month":"04","project":[{"grant_number":"F06504","call_identifier":"FWF","name":"Taming Complexity in Partial Differential Systems","_id":"260482E2-B435-11E9-9278-68D0E5697425"}],"date_updated":"2026-06-04T22:30:22Z","oa":1,"arxiv":1,"department":[{"_id":"GradSch"},{"_id":"JaMa"}],"related_material":{"record":[{"relation":"later_version","status":"public","id":"20865"},{"id":"20563","status":"public","relation":"dissertation_contains"}]},"publication_status":"draft","corr_author":"1","article_processing_charge":"No","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","title":"Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions","publication":"arXiv","article_number":"2403.07803","day":"09","year":"2024","OA_place":"repository","OA_type":"green","language":[{"iso":"eng"}],"date_created":"2025-10-28T13:12:56Z"}]