{"paper":{"title":"The principal eigenvalue of a mixed local and nonlocal operator with drift","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA","math.FA"],"primary_cat":"math.AP","authors_text":"Craig Cowan, Mohammad El Smaily, Pierre Aime Feulefack","submitted_at":"2024-06-27T23:32:12Z","abstract_excerpt":"We study the eigenvalue problem involving the mixed local-nonlocal operator $ L:= -\\Delta +(-\\Delta)^{s}+q\\cdot\\nabla$~ in a bounded domain $\\Omega\\subset\\R^N,$ where a Dirichlet condition is posed on $\\R^N\\setminus\\Omega.$ The field $q$ stands for a drift or advection in the medium. We prove the existence of a principal eigenvalue and a principal eigenfunction for $s\\in (0,1/2]$. Moreover, we prove $C^{2,\\alpha}$ regularity, up to the boundary, of the solution to the problem $Lu=f$ when coupled with a Dirichlet condition and $0<s<1/2$. To prove the regularity and the existence of a principal "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.19577","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2406.19577/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}