{"paper":{"title":"Gradient integrability and rigidity results for two-phase conductivities in dimension two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.AP","authors_text":"Marcello Ponsiglione, Mariapia Palombaro, Vincenzo Nesi","submitted_at":"2012-01-25T16:59:45Z","abstract_excerpt":"This paper deals with higher gradient integrability for $\\sigma$-harmonic functions $u$ with discontinuous coefficients $\\sigma$, i.e. weak solutions of $\\div(\\sigma \\nabla u) = 0$.\n  We focus on two-phase conductivities, and study the higher integrability of the corresponding gradient field $|\\nabla u|$. The gradient field and its integrability clearly depend on the geometry, i.e., on the phases arrangement. We find the optimal integrability exponent of the gradient field corresponding to any pair $\\{\\sigma_1,\\sigma_2\\}$ of positive definite matrices, i.e., the worst among all possible microg"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.5324","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"}