---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20865'
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."
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).
article_number: '23'
article_processing_charge: Yes (via OA deal)
article_type: original
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. Calculus of Variations and Partial Differential
Equations. 2026;65(1). doi:10.1007/s00526-025-03193-1
apa: Quattrocchi, F. (2026). Variational structures for the Fokker-Planck equation
with general Dirichlet boundary conditions. Calculus of Variations and Partial
Differential Equations. Springer Nature. https://doi.org/10.1007/s00526-025-03193-1
chicago: Quattrocchi, Filippo. “Variational Structures for the Fokker-Planck Equation
with General Dirichlet Boundary Conditions.” Calculus of Variations and Partial
Differential Equations. Springer Nature, 2026. https://doi.org/10.1007/s00526-025-03193-1.
ieee: F. Quattrocchi, “Variational structures for the Fokker-Planck equation with
general Dirichlet boundary conditions,” Calculus of Variations and Partial
Differential Equations, 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.
mla: Quattrocchi, Filippo. “Variational Structures for the Fokker-Planck Equation
with General Dirichlet Boundary Conditions.” Calculus of Variations and Partial
Differential Equations, vol. 65, no. 1, 23, Springer Nature, 2026, doi:10.1007/s00526-025-03193-1.
short: F. Quattrocchi, Calculus of Variations and Partial Differential Equations
65 (2026).
corr_author: '1'
date_created: 2025-12-29T12:06:26Z
date_published: 2026-01-01T00:00:00Z
date_updated: 2026-04-07T08:37:46Z
day: '01'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s00526-025-03193-1
external_id:
arxiv:
- '2403.07803'
file:
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checksum: 635370d64abaf444f50f5cca60bba1be
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month: '01'
oa: 1
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grant_number: F6504
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publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
eissn:
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issn:
- 0944-2669
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
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relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Variational structures for the Fokker-Planck equation with general Dirichlet
boundary conditions
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: ba8df636-2132-11f1-aed0-ed93e2281fdd
volume: 65
year: '2026'
...
---
OA_place: repository
OA_type: green
_id: '20251'
abstract:
- lang: eng
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.
acknowledgement: We are grateful to Rupert Frank and Enno Lenzmann for helpful discussions.
article_number: '226'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Eric
full_name: Carlen, Eric
last_name: Carlen
- first_name: Mathieu
full_name: Lewin, Mathieu
last_name: Lewin
- first_name: Elliott H.
full_name: Lieb, Elliott H.
last_name: Lieb
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: Carlen E, Lewin M, Lieb EH, Seiringer R. Stability estimate for the Lane–Emden
inequality. Calculus of Variations and Partial Differential Equations.
2025;64(7). doi:10.1007/s00526-025-03062-x
apa: Carlen, E., Lewin, M., Lieb, E. H., & Seiringer, R. (2025). Stability estimate
for the Lane–Emden inequality. Calculus of Variations and Partial Differential
Equations. Springer Nature. https://doi.org/10.1007/s00526-025-03062-x
chicago: Carlen, Eric, Mathieu Lewin, Elliott H. Lieb, and Robert Seiringer. “Stability
Estimate for the Lane–Emden Inequality.” Calculus of Variations and Partial
Differential Equations. Springer Nature, 2025. https://doi.org/10.1007/s00526-025-03062-x.
ieee: E. Carlen, M. Lewin, E. H. Lieb, and R. Seiringer, “Stability estimate for
the Lane–Emden inequality,” Calculus of Variations and Partial Differential
Equations, vol. 64, no. 7. Springer Nature, 2025.
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.
mla: Carlen, Eric, et al. “Stability Estimate for the Lane–Emden Inequality.” Calculus
of Variations and Partial Differential Equations, vol. 64, no. 7, 226, Springer
Nature, 2025, doi:10.1007/s00526-025-03062-x.
short: E. Carlen, M. Lewin, E.H. Lieb, R. Seiringer, Calculus of Variations and
Partial Differential Equations 64 (2025).
date_created: 2025-08-31T22:01:31Z
date_published: 2025-09-01T00:00:00Z
date_updated: 2025-09-30T14:27:35Z
day: '01'
department:
- _id: RoSe
doi: 10.1007/s00526-025-03062-x
external_id:
arxiv:
- '2410.20113'
isi:
- '001558641300006'
intvolume: ' 64'
isi: 1
issue: '7'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2410.20113
month: '09'
oa: 1
oa_version: Preprint
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: Stability estimate for the Lane–Emden inequality
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 64
year: '2025'
...
---
_id: '15334'
abstract:
- lang: eng
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.
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)."
article_number: '94'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Helmut
full_name: Abels, Helmut
last_name: Abels
- first_name: Mingwen
full_name: Fei, Mingwen
last_name: Fei
- first_name: Maximilian
full_name: Moser, Maximilian
id: a60047a9-da77-11eb-85b4-c4dc385ebb8c
last_name: Moser
citation:
ama: Abels H, Fei M, Moser M. Sharp interface limit for a Navier–Stokes/Allen–Cahn
system in the case of a vanishing mobility. Calculus of Variations and Partial
Differential Equations. 2024;63(4). doi:10.1007/s00526-024-02715-7
apa: 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. Springer Nature. https://doi.org/10.1007/s00526-024-02715-7
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.” Calculus
of Variations and Partial Differential Equations. Springer Nature, 2024. https://doi.org/10.1007/s00526-024-02715-7.
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,” Calculus of Variations and Partial
Differential Equations, vol. 63, no. 4. Springer Nature, 2024.
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.
mla: Abels, Helmut, et al. “Sharp Interface Limit for a Navier–Stokes/Allen–Cahn
System in the Case of a Vanishing Mobility.” Calculus of Variations and Partial
Differential Equations, vol. 63, no. 4, 94, Springer Nature, 2024, doi:10.1007/s00526-024-02715-7.
short: H. Abels, M. Fei, M. Moser, Calculus of Variations and Partial Differential
Equations 63 (2024).
date_created: 2024-04-21T22:00:52Z
date_published: 2024-05-01T00:00:00Z
date_updated: 2025-09-04T13:45:40Z
day: '01'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1007/s00526-024-02715-7
ec_funded: 1
external_id:
arxiv:
- '2304.12096'
isi:
- '001199418100002'
file:
- access_level: open_access
checksum: b1095fad4cae596f52cc616a973bdde2
content_type: application/pdf
creator: dernst
date_created: 2024-04-23T07:30:48Z
date_updated: 2024-04-23T07:30:48Z
file_id: '15343'
file_name: 2024_CalculusEquations_Abels.pdf
file_size: 975186
relation: main_file
success: 1
file_date_updated: 2024-04-23T07:30:48Z
has_accepted_license: '1'
intvolume: ' 63'
isi: 1
issue: '4'
language:
- iso: eng
month: '05'
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: Sharp interface limit for a Navier–Stokes/Allen–Cahn system in the case of
a vanishing mobility
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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 63
year: '2024'
...
---
_id: '17282'
abstract:
- lang: eng
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.
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.
article_number: '153'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Morris
full_name: Brooks, Morris
id: B7ECF9FC-AA38-11E9-AC9A-0930E6697425
last_name: Brooks
orcid: 0000-0002-6249-0928
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
citation:
ama: Brooks M, Maas J. Characterisation of gradient flows for a given functional.
Calculus of Variations and Partial Differential Equations. 2024;63(6).
doi:10.1007/s00526-024-02755-z
apa: Brooks, M., & Maas, J. (2024). Characterisation of gradient flows for a
given functional. Calculus of Variations and Partial Differential Equations.
Springer Nature. https://doi.org/10.1007/s00526-024-02755-z
chicago: Brooks, Morris, and Jan Maas. “Characterisation of Gradient Flows for a
given Functional.” Calculus of Variations and Partial Differential Equations.
Springer Nature, 2024. https://doi.org/10.1007/s00526-024-02755-z.
ieee: M. Brooks and J. Maas, “Characterisation of gradient flows for a given functional,”
Calculus of Variations and Partial Differential Equations, vol. 63, no.
6. Springer Nature, 2024.
ista: Brooks M, Maas J. 2024. Characterisation of gradient flows for a given functional.
Calculus of Variations and Partial Differential Equations. 63(6), 153.
mla: Brooks, Morris, and Jan Maas. “Characterisation of Gradient Flows for a given
Functional.” Calculus of Variations and Partial Differential Equations,
vol. 63, no. 6, 153, Springer Nature, 2024, doi:10.1007/s00526-024-02755-z.
short: M. Brooks, J. Maas, Calculus of Variations and Partial Differential Equations
63 (2024).
corr_author: '1'
date_created: 2024-07-21T22:01:01Z
date_published: 2024-07-01T00:00:00Z
date_updated: 2025-09-08T08:24:51Z
day: '01'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s00526-024-02755-z
ec_funded: 1
external_id:
arxiv:
- '2209.11149'
isi:
- '001258097800003'
pmid:
- '38947856'
file:
- access_level: open_access
checksum: a0cf0e0ba2157aabb287cb597be17dac
content_type: application/pdf
creator: dernst
date_created: 2024-07-22T07:05:32Z
date_updated: 2024-07-22T07:05:32Z
file_id: '17289'
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has_accepted_license: '1'
intvolume: ' 63'
isi: 1
issue: '6'
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
pmid: 1
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
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: Characterisation of gradient flows for a given functional
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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 63
year: '2024'
...
---
_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. Calculus of Variations and Partial Differential
Equations. 2023;62(5). doi:10.1007/s00526-023-02472-z
apa: Gladbach, P., Kopfer, E., Maas, J., & Portinale, L. (2023). Homogenisation
of dynamical optimal transport on periodic graphs. Calculus of Variations and
Partial Differential Equations. Springer Nature. https://doi.org/10.1007/s00526-023-02472-z
chicago: Gladbach, Peter, Eva Kopfer, Jan Maas, and Lorenzo Portinale. “Homogenisation
of Dynamical Optimal Transport on Periodic Graphs.” Calculus of Variations
and Partial Differential Equations. Springer Nature, 2023. https://doi.org/10.1007/s00526-023-02472-z.
ieee: P. Gladbach, E. Kopfer, J. Maas, and L. Portinale, “Homogenisation of dynamical
optimal transport on periodic graphs,” Calculus of Variations and Partial Differential
Equations, 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.” Calculus of Variations and Partial Differential Equations, vol.
62, no. 5, 143, Springer Nature, 2023, doi:10.1007/s00526-023-02472-z.
short: P. Gladbach, E. Kopfer, J. Maas, L. Portinale, Calculus of Variations and
Partial Differential Equations 62 (2023).
corr_author: '1'
date_created: 2023-05-14T22:01:00Z
date_published: 2023-04-28T00:00:00Z
date_updated: 2025-05-15T10:54:12Z
day: '28'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s00526-023-02472-z
ec_funded: 1
external_id:
arxiv:
- '2110.15321'
isi:
- '000980588900001'
pmid:
- '37131846'
file:
- access_level: open_access
checksum: 359bee38d94b7e0aa73925063cb8884d
content_type: application/pdf
creator: dernst
date_created: 2023-10-04T11:34:10Z
date_updated: 2023-10-04T11:34:10Z
file_id: '14393'
file_name: 2023_CalculusEquations_Gladbach.pdf
file_size: 1240995
relation: main_file
success: 1
file_date_updated: 2023-10-04T11:34:10Z
has_accepted_license: '1'
intvolume: ' 62'
isi: 1
issue: '5'
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
pmid: 1
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
grant_number: W1245
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
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
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. Calculus of Variations and Partial
Differential Equations. 2022;61(6). doi:10.1007/s00526-022-02307-3'
apa: '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. Springer Nature. https://doi.org/10.1007/s00526-022-02307-3'
chicago: 'Hensel, Sebastian, and Maximilian Moser. “Convergence Rates for the Allen–Cahn
Equation with Boundary Contact Energy: The Non-Perturbative Regime.” Calculus
of Variations and Partial Differential Equations. Springer Nature, 2022. https://doi.org/10.1007/s00526-022-02307-3.'
ieee: 'S. Hensel and M. Moser, “Convergence rates for the Allen–Cahn equation with
boundary contact energy: The non-perturbative regime,” Calculus of Variations
and Partial Differential Equations, 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.” Calculus
of Variations and Partial Differential Equations, vol. 61, no. 6, 201, Springer
Nature, 2022, doi:10.1007/s00526-022-02307-3.'
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: 2025-04-14T07:53:59Z
day: '24'
ddc:
- '510'
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doi: 10.1007/s00526-022-02307-3
ec_funded: 1
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grant_number: '948819'
name: Bridging Scales in Random Materials
publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
eissn:
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issn:
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publication_status: published
publisher: Springer Nature
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scopus_import: '1'
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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
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...
---
_id: '73'
abstract:
- lang: eng
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.
article_number: '19'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Matthias
full_name: Erbar, Matthias
last_name: Erbar
- first_name: Jan
full_name: Maas, Jan
id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
last_name: Maas
orcid: 0000-0002-0845-1338
- first_name: Melchior
full_name: Wirth, Melchior
last_name: Wirth
citation:
ama: Erbar M, Maas J, Wirth M. On the geometry of geodesics in discrete optimal
transport. Calculus of Variations and Partial Differential Equations. 2019;58(1).
doi:10.1007/s00526-018-1456-1
apa: Erbar, M., Maas, J., & Wirth, M. (2019). On the geometry of geodesics in
discrete optimal transport. Calculus of Variations and Partial Differential
Equations. Springer. https://doi.org/10.1007/s00526-018-1456-1
chicago: Erbar, Matthias, Jan Maas, and Melchior Wirth. “On the Geometry of Geodesics
in Discrete Optimal Transport.” Calculus of Variations and Partial Differential
Equations. Springer, 2019. https://doi.org/10.1007/s00526-018-1456-1.
ieee: M. Erbar, J. Maas, and M. Wirth, “On the geometry of geodesics in discrete
optimal transport,” Calculus of Variations and Partial Differential Equations,
vol. 58, no. 1. Springer, 2019.
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.
mla: Erbar, Matthias, et al. “On the Geometry of Geodesics in Discrete Optimal Transport.”
Calculus of Variations and Partial Differential Equations, vol. 58, no.
1, 19, Springer, 2019, doi:10.1007/s00526-018-1456-1.
short: M. Erbar, J. Maas, M. Wirth, Calculus of Variations and Partial Differential
Equations 58 (2019).
date_created: 2018-12-11T11:44:29Z
date_published: 2019-02-01T00:00:00Z
date_updated: 2026-04-16T09:51:42Z
day: '01'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s00526-018-1456-1
ec_funded: 1
external_id:
arxiv:
- '1805.06040'
isi:
- '000452849400001'
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checksum: ba05ac2d69de4c58d2cd338b63512798
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creator: dernst
date_created: 2019-01-28T15:37:11Z
date_updated: 2020-07-14T12:47:55Z
file_id: '5895'
file_name: 2018_Calculus_Erbar.pdf
file_size: 645565
relation: main_file
file_date_updated: 2020-07-14T12:47:55Z
has_accepted_license: '1'
intvolume: ' 58'
isi: 1
issue: '1'
language:
- iso: eng
month: '02'
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: 260482E2-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: F06504
name: Taming Complexity in Partial Differential Systems
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
name: IST Austria Open Access Fund
publication: Calculus of Variations and Partial Differential Equations
publication_identifier:
issn:
- 0944-2669
publication_status: published
publisher: Springer
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the geometry of geodesics in discrete optimal transport
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: ba8df636-2132-11f1-aed0-ed93e2281fdd
volume: 58
year: '2019'
...