{"paper":{"title":"Stationary level surfaces and Liouville-type theorems characterizing hyperplanes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Shigeru Sakaguchi","submitted_at":"2012-02-16T08:21:16Z","abstract_excerpt":"We consider an entire graph $S$ in $\\mathbb R^{N+1}$ of a continuous real function $f$ over $\\mathbb R^{N}$ with $N\\ge 1$. Let $\\Omega$ be an unbounded domain in $\\mathbb R^{N+1}$ with boundary $S$. Consider nonlinear diffusion equations of the form $\\partial_t U= \\Delta \\phi(U)$ containing the heat equation. Let $U$ be the solution of either the initial-boundary value problem over $\\Omega$ where the initial value equals zero and the boundary value equals 1, or the Cauchy problem where the initial data is the characteristic function of the set $\\mathbb R^{N+1}\\setminus \\Omega$. The problem we "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.3528","kind":"arxiv","version":2},"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"}