{"paper":{"title":"Regularity of Lipschitz free boundaries for the thin one-phase problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Daniela De Silva, Ovidiu Savin","submitted_at":"2012-05-08T17:33:59Z","abstract_excerpt":"We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $$\\label{E} E(u,\\Omega) = \\int_\\Omega |\\nabla u|^2 dX + \\mathcal{H}^n(\\{u>0\\} \\cap \\{x_{n+1} = 0\\}), \\quad \\Omega \\subset \\R^{n+1},$$ among all functions $u\\ge 0$ which are fixed on $\\p \\Omega$.\n  We prove that the free boundary $F(u)=\\p_{\\R^n}\\{u>0\\}$ of a minimizer $u$ has locally finite $\\mathcal{H}^{n-1}$ measure and is a $C^{2,\\alpha}$ surface except on a small singular set of Hausdorff dimension $n-3$. We also obtain $C^{2,\\alpha}$ regularity of Lipschitz "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.1755","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"}