{"paper":{"title":"Approximation of Lipschitz functions preserving boundary values","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.FA","authors_text":"Carlos Mudarra, Robert Deville","submitted_at":"2018-10-09T18:39:10Z","abstract_excerpt":"Given an open subset $\\Omega$ of a Banach space and a Lipschitz function $u_0: \\overline{\\Omega} \\to \\mathbb{R},$ we study whether it is possible to approximate $u_0$ uniformly on $\\Omega$ by $C^k$-smooth Lipschitz functions which coincide with $u_0$ on the boundary $\\partial \\Omega$ of $\\Omega$ and have the same Lipschitz constant as $u_0.$ As a consequence, we show that every $1$-Lipschitz function $u_0: \\overline{\\Omega} \\to \\mathbb{R},$ defined on the closure $\\overline{\\Omega}$ of an open subset $\\Omega$ of a finite dimensional normed space of dimension $n \\geq 2$, and such that the Lipsc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.04205","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"}