{"paper":{"title":"Symmetry of solutions of a mean field equation on flat tori","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Amir Moradifam, Changfeng Gui","submitted_at":"2016-05-23T06:40:41Z","abstract_excerpt":"We study symmetry of solutions of the mean field equation \\[ \\Delta u +\\rho(\\frac{Ke^u}{\\int_{T_\\epsilon} Ke^u} -\\frac{1}{|T_\\epsilon|} )=0\\] on the flat torus $T_\\epsilon=[-\\frac{1}{2\\epsilon}, \\frac{1}{2\\epsilon}] \\times [-\\frac{1}{2}, \\frac{1}{2}]$ with $0<\\epsilon \\leq 1$, where $K\\in C^2({T}_\\epsilon)$ is a positive function with $-\\Delta \\ln K \\leq \\frac{\\rho}{|T_\\epsilon|}$ and $\\rho \\leq 8\\pi$. We prove that if $(x_0,y_0)$ is a critical point of the function $u+ln(K)$, then $u$ is evenly symmetric about the lines $x=x_0$ and $y=y_0$, provided $K$ is evenly symmetric about these lines. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06905","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"}