{"issue":"1","month":"01","keyword":["Applied Mathematics","Computational Mathematics","Analysis"],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","article_processing_charge":"No","title":"Convergence of the Allen--Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to 90°","publication_identifier":{"eissn":["1095-7154"],"issn":["0036-1410"]},"volume":54,"oa_version":"Preprint","page":"114-172","date_published":"2022-01-04T00:00:00Z","main_file_link":[{"open_access":"1","url":" https://doi.org/10.48550/arXiv.2105.08434"}],"isi":1,"type":"journal_article","author":[{"first_name":"Helmut","full_name":"Abels, Helmut","last_name":"Abels"},{"full_name":"Moser, Maximilian","first_name":"Maximilian","id":"a60047a9-da77-11eb-85b4-c4dc385ebb8c","last_name":"Moser"}],"publisher":"Society for Industrial and Applied Mathematics","intvolume":"        54","scopus_import":"1","doi":"10.1137/21m1424925","_id":"12305","department":[{"_id":"JuFi"}],"abstract":[{"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.","lang":"eng"}],"date_updated":"2023-08-04T10:34:56Z","language":[{"iso":"eng"}],"day":"04","publication_status":"published","year":"2022","quality_controlled":"1","oa":1,"article_type":"original","publication":"SIAM Journal on Mathematical Analysis","external_id":{"arxiv":["2105.08434"],"isi":["000762768000004"]},"date_created":"2023-01-16T10:07:00Z","status":"public","citation":{"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>","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>","short":"H. Abels, M. Moser, SIAM Journal on Mathematical Analysis 54 (2022) 114–172.","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>.","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>.","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.","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."}}