{"paper":{"title":"A nonautonomous chain rule in $W^{1,p}$ and $BV$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Graziano Crasta, Guido De Philippis, Luigi Ambrosio, Virginia De Cicco","submitted_at":"2015-12-09T12:55:07Z","abstract_excerpt":"In this paper we consider the chain rule formula for compositions $x\\mapsto F(x, u(x))$ in the case when $u$ has a Sobolev or BV regularity and $F(x,z)$ is separately Sobolev, or BV, with respect to $x$ and $C^1$ with respect to $z$. Our results extend to this \"nonautonomous\" case the results known for compositions $x\\mapsto F(u(x))$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.02839","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"}