{"paper":{"title":"Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"A. San Antolin, J. D. Rossi, L. I. Ignat","submitted_at":"2011-11-17T14:44:13Z","abstract_excerpt":"We find lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form $ T(u) = - \\int_{\\rr^d} K(x,y) (u(y)-u(x)) \\, dy$. Here we consider a kernel $K(x,y)=\\psi (y-a(x))+\\psi(x-a(y))$ where $\\psi$ is a bounded, nonnegative function supported in the unit ball and $a$ means a diffeomorphism on $\\rr^d$. A simple example being a linear function $a(x)= Ax$. The upper and lower bounds that we obtain are given in terms of the Jacobian of $a$ and the integral of $\\psi$. Indeed, in the linear case $a(x) = Ax$ we obtain an explicit expression for the first eigenvalue in the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.4114","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"}