---
_id: '14755'
abstract:
- lang: eng
  text: We consider the sharp interface limit for the scalar-valued and vector-valued
    Allen–Cahn equation with homogeneous Neumann boundary condition in a bounded smooth
    domain Ω of arbitrary dimension N ⩾ 2 in the situation when a two-phase diffuse
    interface has developed and intersects the boundary ∂ Ω. The limit problem is
    mean curvature flow with 90°-contact angle and we show convergence in strong norms
    for well-prepared initial data as long as a smooth solution to the limit problem
    exists. To this end we assume that the limit problem has a smooth solution on
    [ 0 , T ] for some time T > 0. Based on the latter we construct suitable curvilinear
    coordinates and set up an asymptotic expansion for the scalar-valued and the vector-valued
    Allen–Cahn equation. In order to estimate the difference of the exact and approximate
    solutions with a Gronwall-type argument, a spectral estimate for the linearized
    Allen–Cahn operator in both cases is required. The latter will be shown in a separate
    paper, cf. (Moser (2021)).
acknowledgement: "The author gratefully acknowledges support through DFG, GRK 1692
  “Curvature,\r\nCycles and Cohomology” during parts of the work."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Maximilian
  full_name: Moser, Maximilian
  id: a60047a9-da77-11eb-85b4-c4dc385ebb8c
  last_name: Moser
citation:
  ama: 'Moser M. Convergence of the scalar- and vector-valued Allen–Cahn equation
    to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence
    result. <i>Asymptotic Analysis</i>. 2023;131(3-4):297-383. doi:<a href="https://doi.org/10.3233/asy-221775">10.3233/asy-221775</a>'
  apa: 'Moser, M. (2023). Convergence of the scalar- and vector-valued Allen–Cahn
    equation to mean curvature flow with 90°-contact angle in higher dimensions, part
    I: Convergence result. <i>Asymptotic Analysis</i>. IOS Press. <a href="https://doi.org/10.3233/asy-221775">https://doi.org/10.3233/asy-221775</a>'
  chicago: 'Moser, Maximilian. “Convergence of the Scalar- and Vector-Valued Allen–Cahn
    Equation to Mean Curvature Flow with 90°-Contact Angle in Higher Dimensions, Part
    I: Convergence Result.” <i>Asymptotic Analysis</i>. IOS Press, 2023. <a href="https://doi.org/10.3233/asy-221775">https://doi.org/10.3233/asy-221775</a>.'
  ieee: 'M. Moser, “Convergence of the scalar- and vector-valued Allen–Cahn equation
    to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence
    result,” <i>Asymptotic Analysis</i>, vol. 131, no. 3–4. IOS Press, pp. 297–383,
    2023.'
  ista: 'Moser M. 2023. Convergence of the scalar- and vector-valued Allen–Cahn equation
    to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence
    result. Asymptotic Analysis. 131(3–4), 297–383.'
  mla: 'Moser, Maximilian. “Convergence of the Scalar- and Vector-Valued Allen–Cahn
    Equation to Mean Curvature Flow with 90°-Contact Angle in Higher Dimensions, Part
    I: Convergence Result.” <i>Asymptotic Analysis</i>, vol. 131, no. 3–4, IOS Press,
    2023, pp. 297–383, doi:<a href="https://doi.org/10.3233/asy-221775">10.3233/asy-221775</a>.'
  short: M. Moser, Asymptotic Analysis 131 (2023) 297–383.
date_created: 2024-01-08T13:13:28Z
date_published: 2023-02-02T00:00:00Z
date_updated: 2024-01-09T09:22:16Z
day: '02'
department:
- _id: JuFi
doi: 10.3233/asy-221775
external_id:
  arxiv:
  - '2105.07100'
intvolume: '       131'
issue: 3-4
keyword:
- General Mathematics
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2105.07100
month: '02'
oa: 1
oa_version: Preprint
page: 297-383
publication: Asymptotic Analysis
publication_identifier:
  eissn:
  - 1875-8576
  issn:
  - 0921-7134
publication_status: published
publisher: IOS Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Convergence of the scalar- and vector-valued Allen–Cahn equation to mean curvature
  flow with 90°-contact angle in higher dimensions, part I: Convergence result'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 131
year: '2023'
...