--- _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. Asymptotic Analysis. 2023;131(3-4):297-383. doi:10.3233/asy-221775' 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. Asymptotic Analysis. IOS Press. https://doi.org/10.3233/asy-221775' 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.” Asymptotic Analysis. IOS Press, 2023. https://doi.org/10.3233/asy-221775.' 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,” Asymptotic Analysis, 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.” Asymptotic Analysis, vol. 131, no. 3–4, IOS Press, 2023, pp. 297–383, doi:10.3233/asy-221775.' 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' ... --- _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: 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' ... --- _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°. SIAM Journal on Mathematical Analysis. 2022;54(1):114-172. doi:10.1137/21m1424925 apa: 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. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/21m1424925 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°.” SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics, 2022. https://doi.org/10.1137/21m1424925. 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°,” SIAM Journal on Mathematical Analysis, 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°.” SIAM Journal on Mathematical Analysis, vol. 54, no. 1, Society for Industrial and Applied Mathematics, 2022, pp. 114–72, doi:10.1137/21m1424925. short: H. Abels, M. Moser, SIAM Journal on Mathematical Analysis 54 (2022) 114–172. date_created: 2023-01-16T10:07:00Z date_published: 2022-01-04T00:00:00Z date_updated: 2023-08-04T10:34:56Z 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' ...