---
_id: '12959'
abstract:
- lang: eng
  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."
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).
article_number: '143'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Peter
  full_name: Gladbach, Peter
  last_name: Gladbach
- first_name: Eva
  full_name: Kopfer, Eva
  last_name: Kopfer
- first_name: Jan
  full_name: Maas, Jan
  id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
  last_name: Maas
  orcid: 0000-0002-0845-1338
- first_name: Lorenzo
  full_name: Portinale, Lorenzo
  id: 30AD2CBC-F248-11E8-B48F-1D18A9856A87
  last_name: Portinale
citation:
  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>.
  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.
  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.
  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).
date_created: 2023-05-14T22:01:00Z
date_published: 2023-04-28T00:00:00Z
date_updated: 2023-10-04T11:34:49Z
day: '28'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s00526-023-02472-z
ec_funded: 1
external_id:
  arxiv:
  - '2110.15321'
  isi:
  - '000980588900001'
file:
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  date_created: 2023-10-04T11:34:10Z
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intvolume: '        62'
isi: 1
issue: '5'
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '716117'
  name: Optimal Transport and Stochastic Dynamics
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
  grant_number: F6504
  name: Taming Complexity in Partial Differential Systems
- _id: 260788DE-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  name: Dissipation and Dispersion in Nonlinear Partial Differential Equations
publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
  eissn:
  - 1432-0835
  issn:
  - 0944-2669
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Homogenisation of dynamical optimal transport on periodic graphs
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type: journal_article
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volume: 62
year: '2023'
...
---
_id: '12079'
abstract:
- lang: eng
  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).
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."
article_number: '201'
article_processing_charge: No
article_type: original
author:
- first_name: Sebastian
  full_name: Hensel, Sebastian
  id: 4D23B7DA-F248-11E8-B48F-1D18A9856A87
  last_name: Hensel
  orcid: 0000-0001-7252-8072
- first_name: Maximilian
  full_name: Moser, Maximilian
  id: a60047a9-da77-11eb-85b4-c4dc385ebb8c
  last_name: Moser
citation:
  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>'
  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>.'
  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.'
  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).
date_created: 2022-09-11T22:01:54Z
date_published: 2022-08-24T00:00:00Z
date_updated: 2023-08-03T13:48:30Z
day: '24'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s00526-022-02307-3
ec_funded: 1
external_id:
  isi:
  - '000844247300008'
file:
- access_level: open_access
  checksum: b2da020ce50440080feedabeab5b09c4
  content_type: application/pdf
  creator: dernst
  date_created: 2023-01-20T08:56:01Z
  date_updated: 2023-01-20T08:56:01Z
  file_id: '12320'
  file_name: 2022_Calculus_Hensel.pdf
  file_size: 1278493
  relation: main_file
  success: 1
file_date_updated: 2023-01-20T08:56:01Z
has_accepted_license: '1'
intvolume: '        61'
isi: 1
issue: '6'
language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
project:
- _id: 0aa76401-070f-11eb-9043-b5bb049fa26d
  call_identifier: H2020
  grant_number: '948819'
  name: Bridging Scales in Random Materials
publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
  eissn:
  - 1432-0835
  issn:
  - 0944-2669
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Convergence rates for the Allen–Cahn equation with boundary contact energy:
  The non-perturbative regime'
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: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 61
year: '2022'
...