{"paper":{"title":"Optimal lower exponent for the higher gradient integrability of solutions to two-phase elliptic equations in two dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mariapia Palombaro, Silvio Fanzon","submitted_at":"2017-03-21T16:17:44Z","abstract_excerpt":"We study the higher gradient integrability of distributional solutions $u$ to the equation $div(\\sigma \\nabla u) = 0$ in dimension two, in the case when the essential range of $\\sigma$ consists of only two elliptic matrices, i.e., $\\sigma\\in\\{\\sigma_1, \\sigma_2\\}$ a.e. in $\\Omega$.\n  In [4], for every pair of elliptic matrices $\\sigma_1$ and $\\sigma_2$, exponents $p_{\\sigma_1,\\sigma_2}\\in(2,+\\infty)$ and $q_{\\sigma_1,\\sigma_2}\\in (1,2)$ have been characterised so that if $u\\in W^{1,q_{\\sigma_1,\\sigma_2}}(\\Omega)$ is solution to the elliptic equation then $\\nabla u\\in L^{p_{\\sigma_1,\\sigma_2}}_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.07298","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"}