{"paper":{"title":"Some Results on the Schiffer's Conjecture in R^2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Jian Deng","submitted_at":"2011-11-30T01:45:40Z","abstract_excerpt":"Let $\\Omega$ be an open, bounded domain in the plane with connected and smooth boundary, and $\\omega$ an eigenfunction of the Neumann Laplacian corresponding to some Neumann eigenvalue $\\mu > 0$. If the boundary value of $\\omega$ is a nonzero constant along the boundary, denoting $0 = \\mu_1(\\Omega) < \\mu_2(\\Omega) <= ...$ the set of all Neumann eigenvalues for the Laplacian on $\\Omega$, we show that 1) if $\\mu < \\mu_8(\\Omega)$; or 2) if $\\Omega$ is strictly convex and centrally symmetric, $\\mu < \\mu_13(\\Omega)$, then $\\Omega$ must be a disk."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.0207","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"}