{"paper":{"title":"Jacobians of $W^{1,p}$ homeomorphisms, case $p=[n/2]$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.CA","authors_text":"Pawe{\\l} Goldstein, Piotr Haj{\\l}asz","submitted_at":"2018-12-31T16:39:04Z","abstract_excerpt":"We investigate a known problem whether a Sobolev homeomorphism between domains in $\\mathbb{R}^n$ can change sign of the Jacobian. The only case that remains open is when $f\\in W^{1,[n/2]}$, $n\\geq 4$. We prove that if $n\\geq 4$, and a sense-preserving homeomorphism $f$ satisfies $f\\in W^{1,[n/2]}$, $f^{-1}\\in W^{1,n-[n/2]-1}$ and either $f$ is H\\\"older continuous on almost all spheres of dimension $[n/2]$, or $f^{-1}$ is H\\\"older continuous on almost all spheres of dimensions $n-[n/2]-1$, then the Jacobian of $f$ is non-negative, $J_f\\geq 0$, almost everywhere. This result is a consequence of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.11888","kind":"arxiv","version":2},"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"}