{"paper":{"title":"Free Boundary Regularity for Almost-Minimizers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guy David, Max Engelstein, Tatiana Toro","submitted_at":"2017-02-21T21:04:22Z","abstract_excerpt":"In this paper we study the free boundary regularity for almost-minimizers of the functional \\begin{equation*} J(u)=\\int_{\\mathcal O} |\\nabla u(x)|^2 +q^2_+(x)\\chi_{\\{u>0\\}}(x) +q^2_-(x)\\chi_{\\{u<0\\}}(x)\\ dx \\end{equation*} where $q_\\pm \\in L^\\infty(\\mathcal O)$. Almost-minimizers satisfy a variational inequality but not a PDE or a monotonicity formula the way minimizers do (see [AC], [ACF], [CJK], [W]). Nevertheless we succeed in proving that, under a non-degeneracy assumption on $q_\\pm$, the free boundary is uniformly rectifiable. Furthermore, when $q_-\\equiv 0$, and $q_+$ is H\\\"older continu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.06580","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"}