{"paper":{"title":"A new discrete monotonicity formula with application to a two-phase free boundary problem in dimension two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aram Karakhanyan, Serena Dipierro","submitted_at":"2015-09-01T13:25:15Z","abstract_excerpt":"We continue the analysis of the two-phase free boundary problems initiated in \\cite{DK}, where we studied the linear growth of minimizers in a Bernoulli type free boundary problem at the non-flat points and the related regularity of free boundary. There, we also defined the functional $$\\phi_p(r,u,x_0)=\\frac1{r^4}\\int_{B_r(x_0)}\\frac{|\\nabla u^+(x)|^p}{|x-x_0|^{N-2}}dx\\int_{B_r(x_0)}\\frac{|\\nabla u^-(x)|^p}{|x-x_0|^{N-2}}dx$$ where $x_0$ is a free boundary point, i.e. $x_0\\in\\partial\\{u>0\\}$ and $u$ is a minimizer of the functional $$J(u):=\\int_{\\Omega}|\\nabla u|^p +\\lambda_+^p\\,\\chi_{\\{u>0\\}}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.00277","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"}