{"paper":{"title":"The First Eigenvalue of the Kohn-Laplace Operator in the Heisenberg Group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Akram Makni, Najoua Gamara","submitted_at":"2016-03-07T21:10:41Z","abstract_excerpt":"In this paper, by extending the notions of harmonic transplantation and harmonic radius in the Heisenberg group, we give an upper bound for the first eigenvalue for the following Dirichlet problem: $$(P_{\\Omega}) \\left\\{ \\begin{array}{lllll} -\\Delta_{\\mathbb{H}^1} u & = & \\lambda u & \\mbox{in} & \\Omega u & = & 0 & \\mbox{on} & \\partial \\Omega, \\end{array} \\right.$$ where $ \\Omega $ is a regular bounded domain of $ \\mathbb{H}^1$ with smooth boundary and $\\Delta_{\\mathbb{H}^1}$ is the Kohn-Laplace operator. Using the results of P.Pansu which give the relation between the volume of $\\Omega$ and th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.02295","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"}