{"paper":{"title":"Mesh-independent a priori bounds for nonlinear elliptic finite difference boundary value problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"A. Verbitsky, P.J. McKenna, W. Reichel","submitted_at":"2014-04-09T07:53:05Z","abstract_excerpt":"In this paper we prove mesh independent a priori $L^\\infty$-bounds for positive solutions of the finite difference boundary value problem $$ -\\Delta_h u = f(x,u) \\mbox{ in } \\Omega_h, \\quad u=0 \\mbox{ on } \\partial\\Omega_h, $$ where $\\Delta_h$ is the finite difference Laplacian and $\\Omega_h$ is a discretized $n$-dimensional box. On one hand this completes a result of [10] on the asympotic symmetry of solutions of finite difference boundary value problems. On the other hand it is a finite difference version of a critical exponent problem studied in [11]. Two main results are given: one for dim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.2386","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"}