Minimisers of the Allen–Cahn equation on hyperbolic graphs

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• 作者：Blaž Mramor
• 关键词：Mathematics Subject Classification58J05 ; 35J20 ; 53C23
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：56
• 期：1
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-0835
• 卷排序：56

文摘

We investigate minimal solutions of the Allen–Cahn equation on a Gromov-hyperbolic graph. Under some natural conditions on the graph, we show the existence of non-constant uniformly-bounded minimal solutions with prescribed asymptotic behaviours. For a phase field model on a hyperbolic graph, such solutions describe energy-minimising steady-state phase transitions that converge towards prescribed phases given by the asymptotic directions on the graph.