{"paper":{"title":"A new symmetry criterion based on the distance function and applications to PDE's","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Graziano Crasta, Ilaria Fragal\\`a","submitted_at":"2012-07-26T17:38:28Z","abstract_excerpt":"We prove that, if $\\Omega\\subset \\mathbb{R}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\\partial \\Omega$ of the function $$\\varphi(y) = \\int_0^{\\lambda(y)} \\prod_{j=1}^{n-1}[1-t \\kappa_j(y)]\\, dt$$ implies that $\\Omega$ is a ball. Here $k_j(y)$ and $\\lambda(y)$ denote respectively the principal curvatures and the cut value of a boundary point $y \\in \\partial \\Omega$. We apply this geometric result to different symmetry questions for PDE's: an overdetermined system of Monge-Kantorovich type equations (which can be viewed as the limit as $p \\to + \\infty$ of Serrin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.6344","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"}