{"paper":{"title":"The square root problem for second order, divergence form operators with mixed boundary conditions on $L^p$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.FA"],"primary_cat":"math.CA","authors_text":"Joachim Rehberg (WIAS), Nadine Badr (ICJ), Pascal Auscher (LM-Orsay), Robert Haller-Dintelmann","submitted_at":"2012-10-02T14:11:35Z","abstract_excerpt":"We show that, under general conditions, the operator $\\bigl (-\\nabla \\cdot \\mu \\nabla +1\\bigr)^{1/2}$ with mixed boundary conditions provides a topological isomorphism between $W^{1,p}_D(\\Omega)$ and $L^p(\\Omega)$, for $p \\in {]1,2[}$ if one presupposes that this isomorphism holds true for $p=2$. The domain $\\Omega$ is assumed to be bounded, the Dirichlet part $D$ of the boundary has to satisfy the well-known Ahlfors-David condition, whilst for the points from $\\overline {\\partial \\Omega \\setminus D}$ the existence of bi-Lipschitzian boundary charts is required."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.0780","kind":"arxiv","version":3},"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"}