{"paper":{"title":"Second order regularity for elliptic and parabolic equations involving $p$-Laplacian via a fundamental inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hongjie Dong, Peng Fa, Yi Ru-Ya Zhang, Yuan Zhou","submitted_at":"2019-08-05T10:22:07Z","abstract_excerpt":"Denote by $\\Delta$ the Laplacian and by $\\Delta_\\infty $ the $\\infty$-Laplacian. A fundamental inequality is proved for the algebraic structure of $\\Delta v\\Delta_\\infty v$: for every $v\\in C^\\infty$, $$\\ | { |D^2vDv|^2} - {\\Delta v \\Delta_\\infty v } -\\frac12[|D^2v|^2-(\\Delta v)^2]|Dv|^2\\ | \\le \\frac{n-2}2 [|D^2v|^2{|Dv|^2}- |D^2vDv|^2 ]. $$ Based on this, we prove the following results:\n  1. For any $p$-harmonic functions $u$, $p\\in(1,2)\\cup(2,\\infty)$, we have $$|Du|^{\\frac{p-\\gamma}2}Du\\in W^{1,2}_{\\rm loc},$$ with $\\gamma<\\min\\{p+\\frac{n-1}{n},3+\\frac{p-1}{n-1}\\}$. As a by-product, when $p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01547","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/1908.01547/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"}