[{"abstract":[{"lang":"eng","text":"We prove the convergence of a modified Jordan–Kinderlehrer–Otto scheme to a solution\r\nto the Fokker–Planck equation in Ω e R^d with general—strictly positive and temporally\r\nconstant—Dirichlet boundary conditions. We work under mild assumptions on the domain,\r\nthe drift, and the initial datum. In the special case where Ω is an interval in R1, we prove\r\nthat such a solution is a gradient flow—curve of maximal slope—within a suitable space of\r\nmeasures, endowed with a modified Wasserstein distance. Our discrete scheme and modified\r\ndistance draw inspiration from contributions by A. Figalli and N. Gigli [J. Math. Pures\r\nAppl. 94, (2010), pp. 107–130], and J. Morales [J. Math. Pures Appl. 112, (2018), pp. 41–88]\r\non an optimal-transport approach to evolution equations with Dirichlet boundary conditions.\r\nSimilarly to these works, we allow the mass to flow from/to the boundary ∂Ω throughout\r\nthe evolution. However, our leading idea is to also keep track of the mass at the boundary\r\nby working with measures defined on the whole closure Ω . The driving functional is a\r\nmodification of the classical relative entropy that also makes use of the information at the\r\nboundary. As an intermediate result, when Ω is an interval in R1, we find a formula for the\r\ndescending slope of this geodesically nonconvex functional."}],"date_updated":"2026-04-07T08:37:46Z","external_id":{"arxiv":["2403.07803"]},"date_created":"2025-12-29T12:06:26Z","title":"Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions","author":[{"full_name":"Quattrocchi, Filippo","id":"3ebd6ba8-edfb-11eb-afb5-91a9745ba308","first_name":"Filippo","last_name":"Quattrocchi","orcid":"0009-0000-9773-1931"}],"project":[{"name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2"}],"article_number":"23","corr_author":"1","OA_type":"hybrid","date_published":"2026-01-01T00:00:00Z","publication":"Calculus of Variations and Partial Differential Equations","file_date_updated":"2026-01-05T12:36:39Z","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","publication_status":"published","publisher":"Springer Nature","ddc":["510"],"quality_controlled":"1","citation":{"short":"F. Quattrocchi, Calculus of Variations and Partial Differential Equations 65 (2026).","mla":"Quattrocchi, Filippo. “Variational Structures for the Fokker-Planck Equation with General Dirichlet Boundary Conditions.” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 65, no. 1, 23, Springer Nature, 2026, doi:<a href=\"https://doi.org/10.1007/s00526-025-03193-1\">10.1007/s00526-025-03193-1</a>.","ieee":"F. Quattrocchi, “Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions,” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 65, no. 1. Springer Nature, 2026.","ista":"Quattrocchi F. 2026. Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. Calculus of Variations and Partial Differential Equations. 65(1), 23.","chicago":"Quattrocchi, Filippo. “Variational Structures for the Fokker-Planck Equation with General Dirichlet Boundary Conditions.” <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature, 2026. <a href=\"https://doi.org/10.1007/s00526-025-03193-1\">https://doi.org/10.1007/s00526-025-03193-1</a>.","apa":"Quattrocchi, F. (2026). Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00526-025-03193-1\">https://doi.org/10.1007/s00526-025-03193-1</a>","ama":"Quattrocchi F. Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. <i>Calculus of Variations and Partial Differential Equations</i>. 2026;65(1). doi:<a href=\"https://doi.org/10.1007/s00526-025-03193-1\">10.1007/s00526-025-03193-1</a>"},"_id":"20865","oa_version":"Published Version","article_processing_charge":"Yes (via OA deal)","day":"01","intvolume":"        65","department":[{"_id":"JaMa"}],"scopus_import":"1","article_type":"original","arxiv":1,"month":"01","doi":"10.1007/s00526-025-03193-1","issue":"1","related_material":{"record":[{"relation":"earlier_version","id":"20571","status":"public"}]},"has_accepted_license":"1","PlanS_conform":"1","language":[{"iso":"eng"}],"file":[{"file_id":"20945","success":1,"creator":"dernst","access_level":"open_access","relation":"main_file","date_created":"2026-01-05T12:36:39Z","file_size":958382,"content_type":"application/pdf","checksum":"635370d64abaf444f50f5cca60bba1be","date_updated":"2026-01-05T12:36:39Z","file_name":"2026_CalculusVariations_Quattrocchi.pdf"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"year":"2026","oa":1,"status":"public","publication_identifier":{"eissn":["1432-0835"],"issn":["0944-2669"]},"volume":65,"OA_place":"publisher","type":"journal_article","acknowledgement":"The author would like to thank Jan Maas for suggesting this project and for many helpful comments, Antonio Agresti, Lorenzo Dello Schiavo and Julian Fischer for several fruitful discussions, Oliver Tse for pointing out the reference [10], and the anonymous reviewer for carefully reading this manuscript and providing valuable suggestions. He also gratefully acknowledges support from the Austrian Science Fund (FWF) project 10.55776/F65.Open access funding provided by Institute of Science and Technology (IST Austria)."},{"doi":"10.1007/s00526-025-03062-x","issue":"7","language":[{"iso":"eng"}],"oa":1,"year":"2025","arxiv":1,"isi":1,"month":"09","type":"journal_article","acknowledgement":"We are grateful to Rupert Frank and Enno Lenzmann for helpful discussions.","status":"public","publication_identifier":{"eissn":["1432-0835"],"issn":["0944-2669"]},"OA_place":"repository","volume":64,"article_number":"226","date_published":"2025-09-01T00:00:00Z","OA_type":"green","publication":"Calculus of Variations and Partial Differential Equations","publication_status":"published","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","abstract":[{"text":"The Lane–Emden inequality controls (math. formular) in terms of the L^1 and L^p norms of p. We provide a remainder estimate for this inequality in terms of a suitable distance of p to the manifold of optimizers.","lang":"eng"}],"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2410.20113","open_access":"1"}],"date_updated":"2025-09-30T14:27:35Z","title":"Stability estimate for the Lane–Emden inequality","author":[{"full_name":"Carlen, Eric","last_name":"Carlen","first_name":"Eric"},{"full_name":"Lewin, Mathieu","last_name":"Lewin","first_name":"Mathieu"},{"full_name":"Lieb, Elliott H.","last_name":"Lieb","first_name":"Elliott H."},{"full_name":"Seiringer, Robert","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87","last_name":"Seiringer","first_name":"Robert","orcid":"0000-0002-6781-0521"}],"external_id":{"arxiv":["2410.20113"],"isi":["001558641300006"]},"date_created":"2025-08-31T22:01:31Z","article_processing_charge":"No","oa_version":"Preprint","_id":"20251","day":"01","department":[{"_id":"RoSe"}],"intvolume":"        64","scopus_import":"1","article_type":"original","publisher":"Springer Nature","quality_controlled":"1","citation":{"apa":"Carlen, E., Lewin, M., Lieb, E. H., &#38; Seiringer, R. (2025). Stability estimate for the Lane–Emden inequality. <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00526-025-03062-x\">https://doi.org/10.1007/s00526-025-03062-x</a>","ama":"Carlen E, Lewin M, Lieb EH, Seiringer R. Stability estimate for the Lane–Emden inequality. <i>Calculus of Variations and Partial Differential Equations</i>. 2025;64(7). doi:<a href=\"https://doi.org/10.1007/s00526-025-03062-x\">10.1007/s00526-025-03062-x</a>","chicago":"Carlen, Eric, Mathieu Lewin, Elliott H. Lieb, and Robert Seiringer. “Stability Estimate for the Lane–Emden Inequality.” <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00526-025-03062-x\">https://doi.org/10.1007/s00526-025-03062-x</a>.","ista":"Carlen E, Lewin M, Lieb EH, Seiringer R. 2025. Stability estimate for the Lane–Emden inequality. Calculus of Variations and Partial Differential Equations. 64(7), 226.","ieee":"E. Carlen, M. Lewin, E. H. Lieb, and R. Seiringer, “Stability estimate for the Lane–Emden inequality,” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 64, no. 7. Springer Nature, 2025.","mla":"Carlen, Eric, et al. “Stability Estimate for the Lane–Emden Inequality.” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 64, no. 7, 226, Springer Nature, 2025, doi:<a href=\"https://doi.org/10.1007/s00526-025-03062-x\">10.1007/s00526-025-03062-x</a>.","short":"E. Carlen, M. Lewin, E.H. Lieb, R. Seiringer, Calculus of Variations and Partial Differential Equations 64 (2025)."}},{"project":[{"name":"Bridging Scales in Random Materials","grant_number":"948819","call_identifier":"H2020","_id":"0aa76401-070f-11eb-9043-b5bb049fa26d"}],"external_id":{"arxiv":["2304.12096"],"isi":["001199418100002"]},"date_created":"2024-04-21T22:00:52Z","title":"Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of a vanishing mobility","author":[{"last_name":"Abels","first_name":"Helmut","full_name":"Abels, Helmut"},{"full_name":"Fei, Mingwen","last_name":"Fei","first_name":"Mingwen"},{"last_name":"Moser","first_name":"Maximilian","full_name":"Moser, Maximilian","id":"a60047a9-da77-11eb-85b4-c4dc385ebb8c"}],"date_updated":"2025-09-04T13:45:40Z","abstract":[{"text":"We consider the sharp interface limit of a Navier-Stokes/Allen Cahn equation in a bounded smooth domain in two space dimensions, in the case of vanishing mobility mε=ε√, where the small parameter ε>0 related to the thickness of the diffuse interface is sent to zero. For well-prepared initial data and sufficiently small times, we rigorously prove convergence to the classical two-phase Navier-Stokes system with surface tension. The idea of the proof is to use asymptotic expansions to construct an approximate solution and to estimate the difference of the exact and approximate solutions with a spectral estimate for the (at the approximate solution) linearized Allen-Cahn operator. In the calculations we use a fractional order ansatz and new ansatz terms in higher orders leading to a suitable ε-scaled and coupled model problem. Moreover, we apply the novel idea of introducing ε-dependent coordinates.","lang":"eng"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"published","file_date_updated":"2024-04-23T07:30:48Z","publication":"Calculus of Variations and Partial Differential Equations","date_published":"2024-05-01T00:00:00Z","article_number":"94","citation":{"ieee":"H. Abels, M. Fei, and M. Moser, “Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of a vanishing mobility,” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 63, no. 4. Springer Nature, 2024.","short":"H. Abels, M. Fei, M. Moser, Calculus of Variations and Partial Differential Equations 63 (2024).","mla":"Abels, Helmut, et al. “Sharp Interface Limit for a Navier–Stokes/Allen–Cahn System in the Case of a Vanishing Mobility.” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 63, no. 4, 94, Springer Nature, 2024, doi:<a href=\"https://doi.org/10.1007/s00526-024-02715-7\">10.1007/s00526-024-02715-7</a>.","ista":"Abels H, Fei M, Moser M. 2024. Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of a vanishing mobility. Calculus of Variations and Partial Differential Equations. 63(4), 94.","chicago":"Abels, Helmut, Mingwen Fei, and Maximilian Moser. “Sharp Interface Limit for a Navier–Stokes/Allen–Cahn System in the Case of a Vanishing Mobility.” <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s00526-024-02715-7\">https://doi.org/10.1007/s00526-024-02715-7</a>.","apa":"Abels, H., Fei, M., &#38; Moser, M. (2024). Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of a vanishing mobility. <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00526-024-02715-7\">https://doi.org/10.1007/s00526-024-02715-7</a>","ama":"Abels H, Fei M, Moser M. Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of a vanishing mobility. <i>Calculus of Variations and Partial Differential Equations</i>. 2024;63(4). doi:<a href=\"https://doi.org/10.1007/s00526-024-02715-7\">10.1007/s00526-024-02715-7</a>"},"quality_controlled":"1","ddc":["510"],"publisher":"Springer Nature","scopus_import":"1","article_type":"original","intvolume":"        63","department":[{"_id":"JuFi"}],"day":"01","_id":"15334","article_processing_charge":"Yes (via OA deal)","oa_version":"Published Version","month":"05","isi":1,"arxiv":1,"year":"2024","oa":1,"file":[{"date_updated":"2024-04-23T07:30:48Z","file_name":"2024_CalculusEquations_Abels.pdf","checksum":"b1095fad4cae596f52cc616a973bdde2","content_type":"application/pdf","file_size":975186,"relation":"main_file","date_created":"2024-04-23T07:30:48Z","creator":"dernst","success":1,"access_level":"open_access","file_id":"15343"}],"language":[{"iso":"eng"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"issue":"4","doi":"10.1007/s00526-024-02715-7","has_accepted_license":"1","volume":63,"publication_identifier":{"eissn":["1432-0835"],"issn":["0944-2669"]},"status":"public","ec_funded":1,"acknowledgement":"Open Access funding enabled and organized by Projekt DEAL.\r\nM. Fei was partially supported by NSF of China under Grant No. 12271004 and Anhui Provincial Funding Project under Grant Nos. gxbjZD2022009 and 2308085J10. Moreover, M. Moser has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No 948819).","type":"journal_article"},{"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"published","pmid":1,"publication":"Calculus of Variations and Partial Differential Equations","file_date_updated":"2024-07-22T07:05:32Z","date_published":"2024-07-01T00:00:00Z","article_number":"153","corr_author":"1","project":[{"call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117","_id":"256E75B8-B435-11E9-9278-68D0E5697425"},{"name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2"}],"date_created":"2024-07-21T22:01:01Z","external_id":{"pmid":["38947856"],"arxiv":["2209.11149"],"isi":["001258097800003"]},"title":"Characterisation of gradient flows for a given functional","author":[{"orcid":"0000-0002-6249-0928","last_name":"Brooks","first_name":"Morris","id":"B7ECF9FC-AA38-11E9-AC9A-0930E6697425","full_name":"Brooks, Morris"},{"first_name":"Jan","last_name":"Maas","orcid":"0000-0002-0845-1338","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","full_name":"Maas, Jan"}],"date_updated":"2025-09-08T08:24:51Z","abstract":[{"text":"Let  X  be a vector field and  Y  be a co-vector field on a smooth manifold  M. Does there exist a smooth Riemannian metric  gαβ  on  M  such that  Yβ=gαβXα ? The main result of this note gives necessary and sufficient conditions for this to be true. As an application of this result we show that a finite-dimensional ergodic Lindblad equation admits a gradient flow structure for the von Neumann relative entropy if and only if the condition of BKM-detailed balance holds.","lang":"eng"}],"article_type":"original","scopus_import":"1","intvolume":"        63","department":[{"_id":"JaMa"}],"day":"01","_id":"17282","oa_version":"Published Version","article_processing_charge":"Yes (via OA deal)","citation":{"ista":"Brooks M, Maas J. 2024. Characterisation of gradient flows for a given functional. Calculus of Variations and Partial Differential Equations. 63(6), 153.","short":"M. Brooks, J. Maas, Calculus of Variations and Partial Differential Equations 63 (2024).","mla":"Brooks, Morris, and Jan Maas. “Characterisation of Gradient Flows for a given Functional.” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 63, no. 6, 153, Springer Nature, 2024, doi:<a href=\"https://doi.org/10.1007/s00526-024-02755-z\">10.1007/s00526-024-02755-z</a>.","ieee":"M. Brooks and J. Maas, “Characterisation of gradient flows for a given functional,” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 63, no. 6. Springer Nature, 2024.","ama":"Brooks M, Maas J. Characterisation of gradient flows for a given functional. <i>Calculus of Variations and Partial Differential Equations</i>. 2024;63(6). doi:<a href=\"https://doi.org/10.1007/s00526-024-02755-z\">10.1007/s00526-024-02755-z</a>","apa":"Brooks, M., &#38; Maas, J. (2024). Characterisation of gradient flows for a given functional. <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00526-024-02755-z\">https://doi.org/10.1007/s00526-024-02755-z</a>","chicago":"Brooks, Morris, and Jan Maas. “Characterisation of Gradient Flows for a given Functional.” <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s00526-024-02755-z\">https://doi.org/10.1007/s00526-024-02755-z</a>."},"quality_controlled":"1","ddc":["510"],"publisher":"Springer Nature","year":"2024","oa":1,"file":[{"file_id":"17289","access_level":"open_access","creator":"dernst","success":1,"date_created":"2024-07-22T07:05:32Z","relation":"main_file","file_name":"2024_CalculusVariations_Brooks.pdf","date_updated":"2024-07-22T07:05:32Z","checksum":"a0cf0e0ba2157aabb287cb597be17dac","file_size":416622,"content_type":"application/pdf"}],"language":[{"iso":"eng"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"issue":"6","doi":"10.1007/s00526-024-02755-z","has_accepted_license":"1","month":"07","isi":1,"arxiv":1,"acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria).J. M. gratefully acknowledges support by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 716117), and by the Austrian Science Fund (FWF), Project SFB F65. We thank the anonymous referee for valuable comments on the paper.","type":"journal_article","volume":63,"publication_identifier":{"eissn":["1432-0835"],"issn":["0944-2669"]},"status":"public","ec_funded":1},{"title":"Homogenisation of dynamical optimal transport on periodic graphs","author":[{"last_name":"Gladbach","first_name":"Peter","full_name":"Gladbach, Peter"},{"full_name":"Kopfer, Eva","last_name":"Kopfer","first_name":"Eva"},{"orcid":"0000-0002-0845-1338","first_name":"Jan","last_name":"Maas","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","full_name":"Maas, Jan"},{"id":"30AD2CBC-F248-11E8-B48F-1D18A9856A87","full_name":"Portinale, Lorenzo","first_name":"Lorenzo","last_name":"Portinale"}],"external_id":{"arxiv":["2110.15321"],"isi":["000980588900001"],"pmid":["37131846"]},"date_created":"2023-05-14T22:01:00Z","project":[{"name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117","call_identifier":"H2020","_id":"256E75B8-B435-11E9-9278-68D0E5697425"},{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","grant_number":"F6504","name":"Taming Complexity in Partial Differential Systems"},{"_id":"260788DE-B435-11E9-9278-68D0E5697425","name":"Dissipation and dispersion in nonlinear partial differential equations","grant_number":"W1245","call_identifier":"FWF"}],"abstract":[{"text":"This paper deals with the large-scale behaviour of dynamical optimal transport on Zd\r\n-periodic graphs with general lower semicontinuous and convex energy densities. Our main contribution is a homogenisation result that describes the effective behaviour of the discrete problems in terms of a continuous optimal transport problem. The effective energy density can be explicitly expressed in terms of a cell formula, which is a finite-dimensional convex programming problem that depends non-trivially on the local geometry of the discrete graph and the discrete energy density. Our homogenisation result is derived from a Γ\r\n-convergence result for action functionals on curves of measures, which we prove under very mild growth conditions on the energy density. We investigate the cell formula in several cases of interest, including finite-volume discretisations of the Wasserstein distance, where non-trivial limiting behaviour occurs.","lang":"eng"}],"date_updated":"2025-05-15T10:54:12Z","publication":"Calculus of Variations and Partial Differential Equations","file_date_updated":"2023-10-04T11:34:10Z","publication_status":"published","pmid":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","corr_author":"1","article_number":"143","date_published":"2023-04-28T00:00:00Z","ddc":["510"],"citation":{"ista":"Gladbach P, Kopfer E, Maas J, Portinale L. 2023. Homogenisation of dynamical optimal transport on periodic graphs. Calculus of Variations and Partial Differential Equations. 62(5), 143.","ieee":"P. Gladbach, E. Kopfer, J. Maas, and L. Portinale, “Homogenisation of dynamical optimal transport on periodic graphs,” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 62, no. 5. Springer Nature, 2023.","mla":"Gladbach, Peter, et al. “Homogenisation of Dynamical Optimal Transport on Periodic Graphs.” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 62, no. 5, 143, Springer Nature, 2023, doi:<a href=\"https://doi.org/10.1007/s00526-023-02472-z\">10.1007/s00526-023-02472-z</a>.","short":"P. Gladbach, E. Kopfer, J. Maas, L. Portinale, Calculus of Variations and Partial Differential Equations 62 (2023).","ama":"Gladbach P, Kopfer E, Maas J, Portinale L. Homogenisation of dynamical optimal transport on periodic graphs. <i>Calculus of Variations and Partial Differential Equations</i>. 2023;62(5). doi:<a href=\"https://doi.org/10.1007/s00526-023-02472-z\">10.1007/s00526-023-02472-z</a>","apa":"Gladbach, P., Kopfer, E., Maas, J., &#38; Portinale, L. (2023). Homogenisation of dynamical optimal transport on periodic graphs. <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00526-023-02472-z\">https://doi.org/10.1007/s00526-023-02472-z</a>","chicago":"Gladbach, Peter, Eva Kopfer, Jan Maas, and Lorenzo Portinale. “Homogenisation of Dynamical Optimal Transport on Periodic Graphs.” <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s00526-023-02472-z\">https://doi.org/10.1007/s00526-023-02472-z</a>."},"quality_controlled":"1","publisher":"Springer Nature","department":[{"_id":"JaMa"}],"intvolume":"        62","article_type":"original","scopus_import":"1","article_processing_charge":"Yes (via OA deal)","oa_version":"Published Version","_id":"12959","day":"28","month":"04","arxiv":1,"isi":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"language":[{"iso":"eng"}],"file":[{"content_type":"application/pdf","checksum":"359bee38d94b7e0aa73925063cb8884d","file_size":1240995,"date_updated":"2023-10-04T11:34:10Z","file_name":"2023_CalculusEquations_Gladbach.pdf","date_created":"2023-10-04T11:34:10Z","relation":"main_file","access_level":"open_access","success":1,"creator":"dernst","file_id":"14393"}],"oa":1,"year":"2023","has_accepted_license":"1","issue":"5","doi":"10.1007/s00526-023-02472-z","volume":62,"ec_funded":1,"status":"public","publication_identifier":{"issn":["0944-2669"],"eissn":["1432-0835"]},"acknowledgement":"J.M. gratefully acknowledges support by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 716117). J.M and L.P. also acknowledge support from the Austrian Science Fund (FWF), grants No F65 and W1245. E.K. gratefully acknowledges support by the German Research Foundation through the Hausdorff Center for Mathematics and the Collaborative Research Center 1060. P.G. is partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—350398276. We thank the anonymous reviewer for the careful reading and for useful suggestions. Open access funding provided by Austrian Science Fund (FWF).","type":"journal_article"},{"issue":"6","doi":"10.1007/s00526-022-02307-3","has_accepted_license":"1","language":[{"iso":"eng"}],"file":[{"file_name":"2022_Calculus_Hensel.pdf","date_updated":"2023-01-20T08:56:01Z","content_type":"application/pdf","file_size":1278493,"checksum":"b2da020ce50440080feedabeab5b09c4","relation":"main_file","date_created":"2023-01-20T08:56:01Z","creator":"dernst","success":1,"access_level":"open_access","file_id":"12320"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"year":"2022","oa":1,"isi":1,"month":"08","type":"journal_article","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.\r\nOpen Access funding enabled and organized by Projekt DEAL.","status":"public","ec_funded":1,"publication_identifier":{"issn":["0944-2669"],"eissn":["1432-0835"]},"volume":61,"article_number":"201","date_published":"2022-08-24T00:00:00Z","publication":"Calculus of Variations and Partial Differential Equations","file_date_updated":"2023-01-20T08:56:01Z","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","publication_status":"published","abstract":[{"text":"We extend the recent rigorous convergence result of Abels and Moser (SIAM J Math Anal 54(1):114–172, 2022. https://doi.org/10.1137/21M1424925) concerning convergence rates for solutions of the Allen–Cahn equation with a nonlinear Robin boundary condition towards evolution by mean curvature flow with constant contact angle. More precisely, in the present work we manage to remove the perturbative assumption on the contact angle being close to 90∘. We establish under usual double-well type assumptions on the potential and for a certain class of boundary energy densities the sub-optimal convergence rate of order ε12 for general contact angles α∈(0,π). For a very specific form of the boundary energy density, we even obtain from our methods a sharp convergence rate of order ε; again for general contact angles α∈(0,π). Our proof deviates from the popular strategy based on rigorous asymptotic expansions and stability estimates for the linearized Allen–Cahn operator. Instead, we follow the recent approach by Fischer et al. (SIAM J Math Anal 52(6):6222–6233, 2020. https://doi.org/10.1137/20M1322182), thus relying on a relative entropy technique. We develop a careful adaptation of their approach in order to encode the constant contact angle condition. In fact, we perform this task at the level of the notion of gradient flow calibrations. This concept was recently introduced in the context of weak-strong uniqueness for multiphase mean curvature flow by Fischer et al. (arXiv:2003.05478v2).","lang":"eng"}],"date_updated":"2025-04-14T07:53:59Z","date_created":"2022-09-11T22:01:54Z","external_id":{"isi":["000844247300008"]},"author":[{"id":"4D23B7DA-F248-11E8-B48F-1D18A9856A87","full_name":"Hensel, Sebastian","orcid":"0000-0001-7252-8072","first_name":"Sebastian","last_name":"Hensel"},{"full_name":"Moser, Maximilian","id":"a60047a9-da77-11eb-85b4-c4dc385ebb8c","first_name":"Maximilian","last_name":"Moser"}],"title":"Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime","project":[{"_id":"0aa76401-070f-11eb-9043-b5bb049fa26d","call_identifier":"H2020","name":"Bridging Scales in Random Materials","grant_number":"948819"}],"_id":"12079","article_processing_charge":"No","oa_version":"Published Version","day":"24","intvolume":"        61","department":[{"_id":"JuFi"}],"article_type":"original","scopus_import":"1","publisher":"Springer Nature","ddc":["510"],"quality_controlled":"1","citation":{"chicago":"Hensel, Sebastian, and Maximilian Moser. “Convergence Rates for the Allen–Cahn Equation with Boundary Contact Energy: The Non-Perturbative Regime.” <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s00526-022-02307-3\">https://doi.org/10.1007/s00526-022-02307-3</a>.","ama":"Hensel S, Moser M. Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime. <i>Calculus of Variations and Partial Differential Equations</i>. 2022;61(6). doi:<a href=\"https://doi.org/10.1007/s00526-022-02307-3\">10.1007/s00526-022-02307-3</a>","apa":"Hensel, S., &#38; Moser, M. (2022). Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime. <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00526-022-02307-3\">https://doi.org/10.1007/s00526-022-02307-3</a>","mla":"Hensel, Sebastian, and Maximilian Moser. “Convergence Rates for the Allen–Cahn Equation with Boundary Contact Energy: The Non-Perturbative Regime.” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 61, no. 6, 201, Springer Nature, 2022, doi:<a href=\"https://doi.org/10.1007/s00526-022-02307-3\">10.1007/s00526-022-02307-3</a>.","short":"S. Hensel, M. Moser, Calculus of Variations and Partial Differential Equations 61 (2022).","ieee":"S. Hensel and M. Moser, “Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime,” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 61, no. 6. Springer Nature, 2022.","ista":"Hensel S, Moser M. 2022. Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime. Calculus of Variations and Partial Differential Equations. 61(6), 201."}},{"project":[{"name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117","call_identifier":"H2020","_id":"256E75B8-B435-11E9-9278-68D0E5697425"},{"_id":"260482E2-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"F06504","name":"Taming Complexity in Partial Differential Systems"},{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"author":[{"first_name":"Matthias","last_name":"Erbar","full_name":"Erbar, Matthias"},{"orcid":"0000-0002-0845-1338","first_name":"Jan","last_name":"Maas","full_name":"Maas, Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Wirth, Melchior","last_name":"Wirth","first_name":"Melchior"}],"title":"On the geometry of geodesics in discrete optimal transport","date_created":"2018-12-11T11:44:29Z","external_id":{"isi":["000452849400001"],"arxiv":["1805.06040"]},"date_updated":"2026-04-16T09:51:42Z","abstract":[{"text":"We consider the space of probability measures on a discrete set X, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset Y⊆X, it is natural to ask whether they can be connected by a constant speed geodesic with support in Y at all times. Our main result answers this question affirmatively, under a suitable geometric condition on Y introduced in this paper. The proof relies on an extension result for subsolutions to discrete Hamilton-Jacobi equations, which is of independent interest.","lang":"eng"}],"publication_status":"published","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","file_date_updated":"2020-07-14T12:47:55Z","publication":"Calculus of Variations and Partial Differential Equations","date_published":"2019-02-01T00:00:00Z","article_number":"19","quality_controlled":"1","citation":{"ama":"Erbar M, Maas J, Wirth M. On the geometry of geodesics in discrete optimal transport. <i>Calculus of Variations and Partial Differential Equations</i>. 2019;58(1). doi:<a href=\"https://doi.org/10.1007/s00526-018-1456-1\">10.1007/s00526-018-1456-1</a>","apa":"Erbar, M., Maas, J., &#38; Wirth, M. (2019). On the geometry of geodesics in discrete optimal transport. <i>Calculus of Variations and Partial Differential Equations</i>. Springer. <a href=\"https://doi.org/10.1007/s00526-018-1456-1\">https://doi.org/10.1007/s00526-018-1456-1</a>","chicago":"Erbar, Matthias, Jan Maas, and Melchior Wirth. “On the Geometry of Geodesics in Discrete Optimal Transport.” <i>Calculus of Variations and Partial Differential Equations</i>. Springer, 2019. <a href=\"https://doi.org/10.1007/s00526-018-1456-1\">https://doi.org/10.1007/s00526-018-1456-1</a>.","ista":"Erbar M, Maas J, Wirth M. 2019. On the geometry of geodesics in discrete optimal transport. Calculus of Variations and Partial Differential Equations. 58(1), 19.","ieee":"M. Erbar, J. Maas, and M. Wirth, “On the geometry of geodesics in discrete optimal transport,” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 58, no. 1. Springer, 2019.","short":"M. Erbar, J. Maas, M. Wirth, Calculus of Variations and Partial Differential Equations 58 (2019).","mla":"Erbar, Matthias, et al. “On the Geometry of Geodesics in Discrete Optimal Transport.” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 58, no. 1, 19, Springer, 2019, doi:<a href=\"https://doi.org/10.1007/s00526-018-1456-1\">10.1007/s00526-018-1456-1</a>."},"ddc":["510"],"publisher":"Springer","article_type":"original","scopus_import":"1","department":[{"_id":"JaMa"}],"intvolume":"        58","day":"01","oa_version":"Published Version","article_processing_charge":"Yes (via OA deal)","_id":"73","month":"02","arxiv":1,"isi":1,"oa":1,"year":"2019","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"file":[{"file_name":"2018_Calculus_Erbar.pdf","date_updated":"2020-07-14T12:47:55Z","file_size":645565,"checksum":"ba05ac2d69de4c58d2cd338b63512798","content_type":"application/pdf","relation":"main_file","date_created":"2019-01-28T15:37:11Z","creator":"dernst","access_level":"open_access","file_id":"5895"}],"language":[{"iso":"eng"}],"has_accepted_license":"1","issue":"1","doi":"10.1007/s00526-018-1456-1","volume":58,"publication_identifier":{"issn":["0944-2669"]},"ec_funded":1,"status":"public","type":"journal_article"}]