{"paper":{"title":"On a stiff problem in two-dimensional space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Liping Li, Wenjie Sun","submitted_at":"2021-07-17T14:13:12Z","abstract_excerpt":"In this paper we will study a stiff problem in two-dimensional space and especially its probabilistic counterpart. Roughly speaking, the heat equation with a parameter $\\varepsilon>0$ is under consideration: \\[ \\partial_t u^\\varepsilon(t,x)=\\frac{1}{2}\\nabla \\cdot \\left(\\mathbf{A}_\\varepsilon(x)\\nabla u^\\varepsilon(t,x) \\right),\\quad t\\geq 0, x\\in \\mathbb{R}^2, \\] where $\\mathbf{A}_\\varepsilon(x)=\\text{Id}_2$, the identity matrix, for $x\\notin \\Omega_\\varepsilon:=\\{x=(x_1,x_2)\\in \\mathbb{R}^2: |x_2|<\\varepsilon\\}$ while $$\\mathbf{A}_\\varepsilon(x):=\\begin{pmatrix} a_\\varepsilon^- & 0 \\\\ 0 & a^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.08242","kind":"arxiv","version":2},"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/2107.08242/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"}