{"paper":{"title":"Serrin's Overdetermined Problem and Constant Mean Curvature Surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Frank Pacard, Juncheng Wei, Manuel del Pino","submitted_at":"2013-10-16T21:57:04Z","abstract_excerpt":"For all $N \\geq 9$, we find smooth entire epigraphs in $\\R^N$, namely smooth domains of the form $\\Omega : = \\{x\\in \\R^N\\ / \\ x_N > F (x_1,\\ldots, x_{N-1})\\}$, which are not half-spaces and in which a problem of the form\n  $\\Delta u + f(u) = 0 $ in $\\Omega$ has a positive, bounded solution with 0 Dirichlet boundary data and constant Neumann boundary data on $\\partial \\Omega$. This answers negatively for large dimensions a question by Berestycki, Caffarelli and Nirenberg \\cite{bcn2}. In 1971, Serrin \\cite{serrin} proved that a bounded domain where such an overdetermined problem is solvable must"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.4528","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"}