{"paper":{"title":"Schauder estimates for degenerate Monge-Amp\\`ere equations and smoothness of the eigenfunctions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nam Q. Le, Ovidiu Savin","submitted_at":"2015-04-03T19:24:43Z","abstract_excerpt":"We obtain $C^{2,\\beta}$ estimates up to the boundary for solutions to degenerate Monge-Amp\\`ere equations of the type $$ \\det D^2 u = f~~\\text{in}~\\Omega, \\quad \\quad ~f\\sim \\text{dist}^{\\alpha}(\\cdot, \\partial\\Omega)~\\text{near}~\\partial\\Omega,~\\alpha>0. $$ As a consequence we obtain global $C^\\infty$ estimates up to the boundary for the eigenfunctions of the Monge-Amp\\`ere operator $(\\det D^2 u)^{1/n}$ on smooth, bounded, uniformly convex domains in $R^n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.00912","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"}