{"paper":{"title":"Bi-Sobolev homeomorphisms $f$ with $Df$ and $Df^{-1}$ of low rank using laminates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Marcos Oliva","submitted_at":"2016-07-11T14:26:32Z","abstract_excerpt":"Let $\\Omega\\subset \\mathbb{R}^{n}$ be a bounded open set. Given $1\\leq m_1,m_2\\leq n-2$, we construct a homeomorphism $f :\\Omega\\to \\Omega$ that is H\\\"older continuous, $f$ is the identity on $\\partial \\Omega$, the derivative $D f$ has rank $m_1$ a.e.\\ in $\\Omega$, the derivative $D f^{-1}$ of the inverse has rank $m_2$ a.e.\\ in $\\Omega$, $Df\\in W^{1,p}$ and $Df^{-1}\\in W^{1,q}$ for $p<\\min\\{m_1+1,n-m_2\\}$, $q<\\min\\{m_2+1,n-m_1\\}$. The proof is based on convex integration and laminates. We also show that the integrability of the function and the inverse is sharp."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.02972","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"}