{"paper":{"title":"Layer potentials for elliptic operators with DMO-type coefficients: big pieces $Tb$ theorem, quantitative rectifiability, and free boundary problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Andrea Merlo, Carmelo Puliatti, Mihalis Mourgoglou","submitted_at":"2025-05-29T14:28:01Z","abstract_excerpt":"For $n \\geq 2$, we consider the operator $L_A = -\\mathrm{div }(A(\\cdot)\\nabla)$, where $A$ is a uniformly elliptic $(n+1)\\times(n+1)$ matrix with variable coefficients, a Radon measure $\\mu$ on $\\mathbb{R}^{n+1}$, and the associated gradient of the single layer potential operator $T_\\mu$. Under a Dini-type assumption on the mean oscillation of the matrix $A$, we establish the following results:\n  1) A rectifiability criterion for $\\mu$ in terms of $T_\\mu$. Under quantitative geometric and analytic assumptions within a ball $B$ -- including an upper $n$-growth condition on $\\mu$ in $B$, a thin "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.23478","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.23478/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}