{"paper":{"title":"On a long range segregation model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Luis A. Caffarelli, Stefania Patrizi, Veronica Quitalo","submitted_at":"2015-05-20T15:55:44Z","abstract_excerpt":"In this work we study the properties of segregation processes modeled by a family of equations $$ L(u_i) (x) = u_i(x)\\: F_i (u_1, \\ldots, u_K)(x)\\qquad i=1,\\ldots, K $$ where $F_i (u_1, \\ldots, u_K)(x)$ is a non-local factor that takes into consideration the values of the functions $u_j$'s in a full neighborhood of $x.$ We consider as a model problem $$\\Delta u_i^\\ep (x) = \\frac1{\\ep^2} u_i^\\ep (x)\\sum_{i\\neq j} H(u_j^\\ep)(x)$$ where $\\ep$ is a small parameter and $H(u_j^\\ep)(x)$ is for instance $$H(u_j^\\ep)(x)= \\int_{\\mathcal{B}_1 (x)} u_j^\\ep (y)\\, \\text{d}y$$ or $$H(u_j^\\ep)(x)= \\sup_{y\\in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.05433","kind":"arxiv","version":3},"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"}