---
_id: '12305'
abstract:
- lang: eng
  text: This paper is concerned with the sharp interface limit for the Allen--Cahn
    equation with a nonlinear Robin boundary condition in a bounded smooth domain
    Ω⊂\R2. We assume that a diffuse interface already has developed and that it is
    in contact with the boundary ∂Ω. The boundary condition is designed in such a
    way that the limit problem is given by the mean curvature flow with constant α-contact
    angle. For α close to 90° we prove a local in time convergence result for well-prepared
    initial data for times when a smooth solution to the limit problem exists. Based
    on the latter we construct a suitable curvilinear coordinate system and carry
    out a rigorous asymptotic expansion for the Allen--Cahn equation with the nonlinear
    Robin boundary condition. Moreover, we show a spectral estimate for the corresponding
    linearized Allen--Cahn operator and with its aid we derive strong norm estimates
    for the difference of the exact and approximate solutions using a Gronwall-type
    argument.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Helmut
  full_name: Abels, Helmut
  last_name: Abels
- first_name: Maximilian
  full_name: Moser, Maximilian
  id: a60047a9-da77-11eb-85b4-c4dc385ebb8c
  last_name: Moser
citation:
  ama: Abels H, Moser M. Convergence of the Allen--Cahn equation with a nonlinear
    Robin boundary condition to mean curvature flow with contact angle close to 90°.
    <i>SIAM Journal on Mathematical Analysis</i>. 2022;54(1):114-172. doi:<a href="https://doi.org/10.1137/21m1424925">10.1137/21m1424925</a>
  apa: Abels, H., &#38; Moser, M. (2022). Convergence of the Allen--Cahn equation
    with a nonlinear Robin boundary condition to mean curvature flow with contact
    angle close to 90°. <i>SIAM Journal on Mathematical Analysis</i>. Society for
    Industrial and Applied Mathematics. <a href="https://doi.org/10.1137/21m1424925">https://doi.org/10.1137/21m1424925</a>
  chicago: Abels, Helmut, and Maximilian Moser. “Convergence of the Allen--Cahn Equation
    with a Nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact
    Angle Close to 90°.” <i>SIAM Journal on Mathematical Analysis</i>. Society for
    Industrial and Applied Mathematics, 2022. <a href="https://doi.org/10.1137/21m1424925">https://doi.org/10.1137/21m1424925</a>.
  ieee: H. Abels and M. Moser, “Convergence of the Allen--Cahn equation with a nonlinear
    Robin boundary condition to mean curvature flow with contact angle close to 90°,”
    <i>SIAM Journal on Mathematical Analysis</i>, vol. 54, no. 1. Society for Industrial
    and Applied Mathematics, pp. 114–172, 2022.
  ista: Abels H, Moser M. 2022. Convergence of the Allen--Cahn equation with a nonlinear
    Robin boundary condition to mean curvature flow with contact angle close to 90°.
    SIAM Journal on Mathematical Analysis. 54(1), 114–172.
  mla: Abels, Helmut, and Maximilian Moser. “Convergence of the Allen--Cahn Equation
    with a Nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact
    Angle Close to 90°.” <i>SIAM Journal on Mathematical Analysis</i>, vol. 54, no.
    1, Society for Industrial and Applied Mathematics, 2022, pp. 114–72, doi:<a href="https://doi.org/10.1137/21m1424925">10.1137/21m1424925</a>.
  short: H. Abels, M. Moser, SIAM Journal on Mathematical Analysis 54 (2022) 114–172.
corr_author: '1'
date_created: 2023-01-16T10:07:00Z
date_published: 2022-01-04T00:00:00Z
date_updated: 2024-10-09T21:03:58Z
day: '04'
department:
- _id: JuFi
doi: 10.1137/21m1424925
external_id:
  arxiv:
  - '2105.08434'
  isi:
  - '000762768000004'
intvolume: '        54'
isi: 1
issue: '1'
keyword:
- Applied Mathematics
- Computational Mathematics
- Analysis
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: ' https://doi.org/10.48550/arXiv.2105.08434'
month: '01'
oa: 1
oa_version: Preprint
page: 114-172
publication: SIAM Journal on Mathematical Analysis
publication_identifier:
  eissn:
  - 1095-7154
  issn:
  - 0036-1410
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Convergence of the Allen--Cahn equation with a nonlinear Robin boundary condition
  to mean curvature flow with contact angle close to 90°
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 54
year: '2022'
...