{"paper":{"title":"Minimal Lipschitz and $\\infty$-Harmonic Extensions of Vector-Valued Functions on Finite Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Gabriele Steidl, Johannes Hertrich, Miroslav Ba\\v{c}\\'ak, Sebastian Neumayer","submitted_at":"2019-03-12T12:46:32Z","abstract_excerpt":"This paper deals with extensions of vector-valued functions on finite graphs fulfilling distinguished minimality properties. We show that so-called lex and L-lex minimal extensions are actually the same and call them minimal Lipschitz extensions. Then we prove that the solution of the graph $p$-Laplacians converge to these extensions as $p\\to \\infty$. Furthermore, we examine the relation between minimal Lipschitz extensions and iterated weighted midrange filters and address their connection to $\\infty$-Laplacians for scalar-valued functions. A convergence proof for an iterative algorithm propo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.04873","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"}