2023/12/18 by Francesco Maggi, Maggi, Francesco, Michael Novack +3 · 2 citations
Earth and Planetary Sciences · Materials Science · Physics and Astronomy · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Solidification and crystal growth phenomena #Theoretical and Computational Physics #nanoparticles nucleation surface interactions
paper · pdf · doi:10.48550/arxiv.2312.11139
openalex publication_date 2023/12/18 · openalex created_date 2023/12/20 · openalex updated_date 2026/07/28
We introduce a diffused interface formulation of the Plateau problem, where the Allen--Cahn energy ACε 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 ε→ 0+, we prove convergence to the Gauss' capillarity formulation of the Plateau problem with positive volume v; and when ε→ 0+, v→ 0+ and ε/v→ 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.