{"paper":{"title":"Expansion of Green's function and regularity of Robin's function for elliptic operators in divergence form","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Daomin Cao, Jie Wan","submitted_at":"2024-01-21T13:18:17Z","abstract_excerpt":"We consider Green's function $ G_K $ of the elliptic operator in divergence form $ \\mathcal{L}_K=-\\text{div}(K(x)\\nabla ) $ on a bounded smooth domain $ \\Omega\\subseteq\\mathbb{R}^n (n\\geq 2) $ with zero Dirichlet boundary condition, where $ K $ is a smooth positively definite matrix-valued function on $ \\Omega $. We obtain a high-order asymptotic expansion of $ G_K(x, y) $, which defines uniquely a regular part $ H_K(x, y) $. Moreover, we prove that the associated Robin's function $ R_K(x) = H_K(x, x) $ is smooth in $ \\Omega $, despite the regular part $ H_K\\notin C^1(\\Omega\\times\\Omega) $ in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.11486","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2401.11486/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}