{"paper":{"title":"Cavity problems in discontinuous media","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Disson dos Prazeres, Eduardo V. Teixeira","submitted_at":"2015-12-07T11:59:07Z","abstract_excerpt":"We study cavitation type equations, $\\text{div}(a_{ij}(X) \\nabla u) \\sim \\delta_0(u)$, for bounded, measurable elliptic media $a_{ij}(X)$. De Giorgi-Nash-Moser theory assures that solutions are $\\alpha$-H\\\"older continuous within its set of positivity, $\\{u>0\\}$, for some exponent $\\alpha$ strictly less than one. Notwithstanding, the key, main result proven in this paper provides a sharp Lipschitz regularity estimate for such solutions along their free boundaries, $\\partial \\{u>0 \\}$. Such a sharp estimate implies geometric-measure constrains for the free boundary. In particular, we show that "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.02002","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"}