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On the Gamma -convergence of the Allen-Cahn functional with boundary conditions

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arxiv 2301.07458 v2 pith:36RH3T3J submitted 2023-01-18 math.AP

On the Gamma -convergence of the Allen-Cahn functional with boundary conditions

classification math.AP
keywords functionalminimizersvarepsilonallen-cahnenergygammalimitlimiting
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We study minimizers of the Allen-Cahn system. We consider the $ \varepsilon $-energy functional with Dirichlet values and we establish the $ \Gamma $-limit. The minimizers of the limiting functional are closely related to minimizing partitions of the domain. Finally, utilizing that the triod and the straight line are the only minimal cones in the plane together with regularity results for minimal curves, we determine the precise structure of the minimizers of the limiting functional, and thus the limit of minimizers of the $ \varepsilon $-energy functional as $ \varepsilon \rightarrow 0 $.

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