{"paper":{"title":"On partially segregated harmonic maps: optimal regularity and structure of the free boundary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nicola Soave, Susanna Terracini","submitted_at":"2024-10-31T14:30:35Z","abstract_excerpt":"We consider triplets of densities $(u_1,u_2,u_3)$ minimizing the Dirichlet energy \\[\\sum_{j=1}^3 \\int_{\\Omega} |\\nabla u_j|^2\\,dx \\] over a bounded domain $\\Omega\\subset \\mathbb{R}^N$, subject to the partial segregation condition: \\[ u_1\\,u_2\\,u_3 \\equiv 0 \\ \\text{in $\\Omega$.} \\] We prove optimal regularity of the minimizers in spaces of H\\\"older continuous functions of exponent $3/4$; furthermore we prove that the free boundary is a collection of a locally finite number of smooth codimension one manifolds up to a residual set of Hausdorff dimension at most $N-2$. Finally we prove uniform-in-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.23976","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/2410.23976/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"}