{"paper":{"title":"Bi-Sobolev Solutions to the Prescribed Jacobian Inequality in the Plane with $L^p$ Data","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Julian Fischer, Olivier Kneuss","submitted_at":"2014-08-07T13:45:24Z","abstract_excerpt":"We construct planar bi-Sobolev mappings whose local volume distortion is bounded from below by a given function $f\\in L^p$ with $p>1$, i.e. bi-Sobolev solutions for the prescribed Jacobian inequality in the plane for right-hand sides $f\\in L^p$. More precisely, for any $1<q<(p+1)/2$ we construct a solution which belongs to $W^{1,q}$ and which preserves the boundary pointwise. For bounded right-hand sides $f\\in L^{\\infty}$, we provide bi-Lipschitz solutions. The basic building block of our construction are Lipschitz maps which stretch a given compact subset of the unit square by a given factor "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.1587","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"}