{"paper":{"title":"Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $\\ell$ simply connected manifolds when $s \\ge 1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.FA","authors_text":"Augusto C. Ponce, Jean Van Schaftingen, Pierre Bousquet","submitted_at":"2012-10-09T08:18:00Z","abstract_excerpt":"Given a compact manifold $N^n \\subset \\mathbb{R}^\\nu$, $s \\ge 1$ and $1 \\le p < \\infty$, we prove that the class of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s, p}(Q^m; N^n)$ when $N^n$ is $\\lfloor sp \\rfloor$ simply connected. For $sp$ integer, we prove weak density of smooth maps with values into $N^n$ when $N^n$ is $sp - 1$ simply connected. The proofs are based on the existence of a retraction of $\\mathbb{R}^\\nu$ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.2525","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"}