{"paper":{"title":"Measure Upper Bounds for Nodal Sets of Eigenfunctions of the bi-Harmonic Operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Long Tian, Xiaoping Yang","submitted_at":"2017-09-01T04:40:45Z","abstract_excerpt":"In this article, we consider eigenfunctions $u$ of the bi-harmonic operator, i.e.,\n  $\\triangle^2u=\\lambda^2u$ on $\\Omega$ with some homogeneous linear boundary conditions. We assume that $\\Omega\\subseteq\\mathbb{R}^n$ ($n\\geq2$) is a $C^{\\infty}$ bounded domain, $\\partial\\Omega$ is piecewise analytic and $\\partial\\Omega$ is analytic except a set $\\Gamma\\subseteq\\partial\\Omega$ which is a finite union of some compact $(n-2)$ dimensional submanifolds of $\\partial\\Omega$. The main result of this paper is that the measure upper bounds of the nodal sets of the eigenfunctions is controlled by $\\sqrt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.00153","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"}