{"paper":{"title":"On the impossibility of $W_p^2$ estimates for elliptic equations with piecewise constant coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Doyoon Kim, Hongjie Dong","submitted_at":"2014-04-22T20:58:48Z","abstract_excerpt":"In this paper, we present counterexamples showing that for any $p\\in (1,\\infty)$, $p\\neq 2$, there is a non-divergence form uniformly elliptic operator with piecewise constant coefficients in $\\mathbb{R}^2$ (constant on each quadrant in $\\mathbb{R}^2$) for which there is no $W^2_p$ estimate. The corresponding examples in the divergence case are also discussed. One implication of these examples is that the ranges of $p$ are sharp in the recent results obtained in [4,5] for non-divergence type elliptic and parabolic equations in a half space with the Dirichlet or Neumann boundary condition when "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.5647","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"}