{"paper":{"title":"Two- and Multi-phase Quadrature Surfaces","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Avetik Arakelyan, Henrik Shahgholian, Jyotshana V. Prajapat","submitted_at":"2016-10-09T07:27:42Z","abstract_excerpt":"In this paper we shall initiate the study of the two- and multi-phase quadrature surfaces (QS), which amounts to a two/multi-phase free boundary problems of Bernoulli type. The problem is studied mostly from a potential theoretic point of view that (for two-phase case) relates to integral representation $$ \\int_{\\partial \\Omega^+} g h (x) \\ d\\sigma_x - \\int_{\\partial \\Omega^-} g h (x) \\ d\\sigma_x= \\int h d\\mu \\ , $$ where $d\\sigma_x$ is the surface measure, $\\mu= \\mu^+ - \\mu^-$ is given measure with support in (a priori unknown domain) $\\Omega$, $g$ is a given smooth positive function, and the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.02637","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"}