{"paper":{"title":"Extrapolation for the $L^p$ Dirichlet Problem in Lipschitz domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Zhongwei Shen","submitted_at":"2018-01-02T20:36:21Z","abstract_excerpt":"Let $\\mathcal{L}$ be a second-order linear elliptic operator with complex coefficients. We show that if the $L^p$ Dirichlet problem for the elliptic system $\\mathcal{L}(u)=0$ in a fixed Lipschitz domain $\\Omega$ in $\\mathbb{R}^d$ is solvable for some $1<p=p_0< \\frac{2(d-1)}{d-2}$, then it is solvable for all $p$ satisfying $$ p_0<p< \\frac{2(d-1)}{d-2} +\\varepsilon. $$ The proof is based on a real-variable argument. It only requires that local solutions of $\\mathcal{L}(u)=0$ satisfy a boundary Cacciopoli inequality."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00828","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"}