Lagrange multiplier and singular limit of double obstacle problems for the Allen–Cahn equation with constraint†
Mohammad Hassan Farshbaf-Shaker
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, Berlin, 10117 Germany
Search for more papers by this authorTakeshi Fukao
Department of Mathematics, Kyoto University of Education, 1 Fujinomori, Fukakusa, Fushimi-ku, Kyoto, 612-8522 Japan
Search for more papers by this authorCorresponding Author
Noriaki Yamazaki
Department of Mathematics, Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama, 221-8686 JAPAN
Correspondence to: Noriaki Yamazaki, Department of Mathematics, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama 221-8686, Japan
E-mail: [email protected]
Search for more papers by this authorMohammad Hassan Farshbaf-Shaker
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, Berlin, 10117 Germany
Search for more papers by this authorTakeshi Fukao
Department of Mathematics, Kyoto University of Education, 1 Fujinomori, Fukakusa, Fushimi-ku, Kyoto, 612-8522 Japan
Search for more papers by this authorCorresponding Author
Noriaki Yamazaki
Department of Mathematics, Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama, 221-8686 JAPAN
Correspondence to: Noriaki Yamazaki, Department of Mathematics, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama 221-8686, Japan
E-mail: [email protected]
Search for more papers by this authorDedicated to Professor Nobuyuki Kenmochi on the occasion of his 70th birthdays
Abstract
We study the properties of the Lagrange multiplier for an Allen–Cahn equation with a double obstacle potential. Here, the dynamic boundary condition, including the Laplace–Beltrami operator on the boundary, is investigated. We then establish the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier of our problem. We present remarks on a trace problem as well as on the Neumann boundary condition. Moreover, we describe a numerical experiment for a problem with Neumann boundary condition using the Lagrange multiplier. Copyright © 2016 John Wiley & Sons, Ltd.
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