{"paper":{"title":"Group Theoretic Constructions of Singular Set in a Long Range Segregation Model","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.AP","authors_text":"Howen Chuah","submitted_at":"2026-06-30T06:10:17Z","abstract_excerpt":"In this paper, we construct several explicit examples of singular sets of Hausdorff dimension $(n-2)$ in $\\mathbb{R}^n$ on free boundaries for an elliptic system modeling long range segregation. The system has been previously studied by Caffarelli, Patrizi and Quitalo in \\cite{CL2} for the regularity of the free boundary in dimension two, and by the author and Torres in \\cite{ChPaTo26_2} for the partial regularity in higher dimensions. However, the dimension of the singular set is unknown, and no concrete examples of singular set are known in the literature due to the nonlocal nature of the el"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.31181","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/2606.31181/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"}