{"paper":{"title":"Spectrum of the Laplacian with weights","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Ahmad El Soufi (LMPT), Bruno Colbois","submitted_at":"2016-06-12T08:27:50Z","abstract_excerpt":"Given a compact Riemannian manifold (M, g) and two positive functions $\\rho$ and $\\sigma$, we are interested in the eigenvalues of the Dirichlet energy functional weighted by $\\sigma$, with respect to the L 2 inner product weighted by $\\rho$. Under some regularity conditions on $\\rho$ and $\\sigma$, these eigenvalues are those of the operator $\\rho$^{-1} div($\\sigma$$\\nabla$u) with Neumann conditions on the boundary if $\\partial$M = $\\emptyset$. We investigate the effect of the weights on eigenvalues and discuss the existence of lower and upper bounds under the condition that the total mass is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.04095","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"}