{"paper":{"title":"Variational problems with long-range interaction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessandro Zilio, Hugo Tavares, Nicola Soave, Susanna Terracini","submitted_at":"2017-01-18T10:52:56Z","abstract_excerpt":"We consider a class of variational problems for densities that repel each other at distance. Typical examples are given by the Dirichlet functional and the Rayleigh functional \\[\n  D(\\mathbf{u}) = \\sum_{i=1}^k \\int_{\\Omega} |\\nabla u_i|^2 \\quad \\text{or} \\quad R(\\mathbf{u}) = \\sum_{i=1}^k \\frac{\\int_{\\Omega} |\\nabla u_i|^2}{\\int_{\\Omega} u_i^2} \\] minimized in the class of $H^1(\\Omega,\\mathbb{R}^k)$ functions attaining some boundary conditions on $\\partial \\Omega$, and subjected to the constraint \\[\n  \\mathrm{dist} (\\{u_i > 0\\}, \\{u_j > 0\\}) \\ge 1 \\qquad \\forall i \\neq j. \\] For these problems"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.05005","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":""},"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"}