{"paper":{"title":"A hierarchy of Plateau problems and the approximation of Plateau's laws via the Allen--Cahn equation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.DG","math.MP"],"primary_cat":"math.AP","authors_text":"Daniel Restrepo, Francesco Maggi, Michael Novack","submitted_at":"2023-12-18T12:30:33Z","abstract_excerpt":"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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2312.11139","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2312.11139/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}