{"paper":{"title":"Estimates for eigenvalues of a system of of elliptic equations and of the biharmonic operator","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Changyu Xia, Daguang Chen, Qiaoling Wang, Qing-Ming Cheng","submitted_at":"2010-05-17T15:29:51Z","abstract_excerpt":"Let $\\om $ be a bounded domain in an $n$-dimensional Euclidean space $\\Bbb R^n$. We study eigenvalues of an eigenvalue problem of a system of elliptic equations:\n  $$  \\{\\aligned &\\Delta {\\mathbf u}+ \\alpha{\\rm grad}(\\text{div}{\\mathbf u})=-\\sigma {\\mathbf u}, \\ \\text{in $\\Omega$},\n  &{\\mathbf u}|_{\\partial \\Omega}={\\mathbf 0}. \\aligned . $$ Estimates for eigenvalues of the above eigenvalue problem are obtained. Furthermore, we obtain an upper bound on the $(k+1)^{\\text{th}}$ eigenvalue $\\sigma_{k+1}$. We also obtain sharp lower bound for the first eigenvalue of two kinds of eigenvalue problem"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.2954","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"}