{"paper":{"title":"Singular measure as principal eigenfunction of some nonlocal operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jerome Coville (BIOSP)","submitted_at":"2013-02-05T07:16:08Z","abstract_excerpt":"In this paper, we are interested in the spectral properties of the generalised principal eigenvalue of some nonlocal operator. That is, we look for the existence of some particular solution $(\\lambda,\\phi)$ of a nonlocal operator. $$\\int_{\\O}K(x,y)\\phi(y)\\, dy +a(x)\\phi(x) =-\\lambda \\phi(x),$$ where $\\O\\subset\\R^n$ is an open bounded connected set, $K$ a nonnegative kernel and $a$ is continuous. We prove that for the generalised principal eigenvalue $\\lambda_p:=\\sup \\{\\lambda \\in \\R \\, |\\, \\exists \\, \\phi \\in C(\\O), \\phi > 0 \\;\\text{so that}\\; \\oplb{\\phi}{\\O}+ a(x)\\phi + \\lambda\\phi\\le 0\\}$ th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.0949","kind":"arxiv","version":2},"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"}