{"paper":{"title":"On the boundary conditions in estimating $\\nabla \\omega$ by div $\\omega$ and curl $\\omega.$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dhanya Rajendran, Gyula Csat\\'o, Olivier Kneuss","submitted_at":"2017-09-18T18:32:21Z","abstract_excerpt":"In this paper we study under what boundary conditions the inequality $$\\|\\nabla\\omega\\|_{L^2(\\Omega)}^2\\leq C\\left(\\|{\\rm curl}\\omega\\|_{L^2(\\Omega)}^2+ \\|{\\rm div}\\omega\\|_{L^2(\\Omega)}^2+\\|\\omega\\|_{L^2(\\Omega)}^2\\right) $$ holds true. It is known that such an estimate holds if either the tangential or normal component of $\\omega$ vanishes on the boundary $\\partial\\omega.$ We show that the vanishing tangential component condition is a special case of a more general one. In two dimensions we give an interpolation result between these two classical boundary conditions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.06117","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"}