{"paper":{"title":"The Dirichlet problem without the maximum principle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"A.F.M. ter Elst, W. Arendt","submitted_at":"2018-03-20T10:44:53Z","abstract_excerpt":"Consider the Dirichlet problem with respect to an elliptic operator \\[ A = - \\sum_{k,l=1}^d \\partial_k \\, a_{kl} \\, \\partial_l\n  - \\sum_{k=1}^d \\partial_k \\, b_k\n  + \\sum_{k=1}^d c_k \\, \\partial_k\n  + c_0 \\] on a bounded Wiener regular open set $\\Omega \\subset R^d$, where $a_{kl}, c_k \\in L_\\infty(\\Omega,R)$ and $b_k,c_0 \\in L_\\infty(\\Omega,C)$. Suppose that the associated operator on $L_2(\\Omega)$ with Dirichlet boundary conditions is invertible. Then we show that for all $\\varphi \\in C(\\partial \\Omega)$ there exists a unique $u \\in C(\\overline \\Omega) \\cap H^1_{\\rm loc}(\\Omega)$ such that $u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.07357","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"}