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arXiv · 2312.11139

A hierarchy of Plateau problems and the approximation of Plateau's laws via the Allen--Cahn equation

Francesco Maggi·Michael Novack·Daniel Restrepo
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Abstract

We introduce a diffused interface formulation of the Plateau problem, where the Allen--Cahn energy $\mathcal{AC}_\varepsilon$ is minimized under a volume constraint $v$ and a spanning condition on the level sets of the densities. We discuss two singular limits of these Allen--Cahn Plateau problems: when $\varepsilon\to 0^+$, we prove convergence to the Gauss' capillarity formulation of the Plateau problem with positive volume $v$; and when $\varepsilon\to 0^+$, $v\to 0^+$ and $\varepsilon/v\to 0^+$, we prove convergence to the classical Plateau problem (in the homotopic spanning formulation of Harrison and Pugh). As a corollary of our analysis we resolve the incompatibility between Plateau's laws and the Allen--Cahn equation implied by a regularity theorem of Tonegawa and Wickramasekera. In particular, we show that Plateau-type singularities can be approximated by energy minimizing solutions of the Allen--Cahn equation with a volume Lagrange multiplier and a transmission condition on a spanning free boundary.

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BibTeXRIS

Francesco Maggi, Michael Novack, Daniel Restrepo. 2023-12-18. A hierarchy of Plateau problems and the approximation of Plateau's laws via the Allen--Cahn equation. https://arxiv.org/abs/2312.11139

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