{"paper":{"title":"Sequential weak continuity of null Lagrangians at the boundary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Agnieszka Kalamajska, Martin Kruzik, Stefan Kroemer","submitted_at":"2012-10-04T14:14:29Z","abstract_excerpt":"We show weak* in measures on $\\bar\\O$/ weak-$L^1$ sequential continuity of $u\\mapsto f(x,\\nabla u):W^{1,p}(\\O;\\R^m)\\to L^1(\\O)$, where $f(x,\\cdot)$ is a null Lagrangian for $x\\in\\O$, it is a null Lagrangian at the boundary for $x\\in\\partial\\O$ and $|f(x,A)|\\le C(1+|A|^p)$. We also give a precise characterization of null Lagrangians at the boundary in arbitrary dimensions. Our results explain, for instance, why $u\\mapsto \\det\\nabla u:W^{1,n}(\\O;\\R^n)\\to L^1(\\O)$ fails to be weakly continuous. Further, we state a new weak lower semicontinuity theorem for integrands depending on null Lagrangians "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.1454","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"}