{"paper":{"title":"Extensions and their Minimizations on the Sierpinski Gasket","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Baris Evren Ugurcan, Nicholas Ryder, Pak Hin Li, Robert S. Strichartz","submitted_at":"2013-08-30T18:14:35Z","abstract_excerpt":"We study the extension problem on the Sierpinski Gasket ($SG$). In the first part we consider minimizing the functional $\\mathcal{E}_{\\lambda}(f) = \\mathcal{E}(f,f) + \\lambda \\int f^2 d \\mu$ with prescribed values at a finite set of points where $\\mathcal{E}$ denotes the energy (the analog of $\\int |\\nabla f|^2$ in Euclidean space) and $\\mu$ denotes the standard self-similiar measure on $SG$. We explicitly construct the minimizer $f(x) = \\sum_{i} c_i G_{\\lambda}(x_i, x)$ for some constants $c_i$, where $G_{\\lambda}$ is the resolvent for $\\lambda \\geq 0$. We minimize the energy over sets in $SG"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.6814","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"}