{"paper":{"title":"Regularity of the correctors and local gradient estimate of the homogenization for the elliptic equation: linear periodic case","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"JunZhi Cui, QiaoFu Zhang","submitted_at":"2011-09-06T08:18:36Z","abstract_excerpt":"$C^\\alpha$ and $W^{1,\\infty}$ estimates for the first-order and second-order correctors in the homogenization are presented based on the translation invariant and Li-Vogelius's gradient estimate for the second order linear elliptic equation with piecewise smooth coefficients. If the data are smooth enough, the error of the first-order expansion for piecewise smooth coefficients is locally $O(\\epsilon)$ in the H\\\"older norm; it is locally $O(\\epsilon)$ in $W^{1,\\infty}$ when coefficients are Lipschitz continuous. It can be partly extended to the nonlinear parabolic equation."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.1107","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"}