{"paper":{"title":"A nonlocal free boundary problem with Wasserstein distance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aram Karakhanyan","submitted_at":"2019-04-12T15:21:11Z","abstract_excerpt":"We study the probability measures $\\rho\\in \\mathcal M(\\mathbb R^2)$ minimizing the functional \\[ J[\\rho]=\\iint \\log\\frac1{|x-y|}d\\rho(x)d\\rho(y)+d^2(\\rho, \\rho_0), \\] where $\\rho_0$ is a given probability measure and $d(\\rho, \\rho_0)$ is the 2-Wasserstein distance of $\\rho$ and $\\rho_0$. %\n  We prove the existence of minimizers $\\rho$ and show that the potential $U^\\rho=-\\log|x|\\ast \\rho$ solves a degenerate obstacle problem, the obstacle being the transport potential. Every minimizer $\\rho$ is absolutely continuous with respect to the Lebesgue measure. The singular set of the free boundary of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.06270","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"}