{"paper":{"title":"Higher Regularity of Homogeneous Gradient Compositions for $p$-Laplace-Type Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Le Xuan Truong, Quoc Hung Nguyen","submitted_at":"2026-08-12T02:53:45Z","abstract_excerpt":"In this paper, we study higher regularity of homogeneous functions of the gradient of solutions to the inhomogeneous $p$-Laplace equation $\\operatorname{div}(|Du|^{p-2}Du)=f$. Although a solution need not be of class $C^2$ across its critical set, its gradient is locally H\\\"older continuous. Suppose that $Du\\in C^{0,\\alpha}_{\\rm loc}$ with $\\alpha\\le 1/(p-1)$, and let $\\Phi$ be smooth away from the origin and positively homogeneous of degree $m$. We prove that $\\Phi(Du)\\in C^k_{\\rm loc}$ whenever $m>k/\\alpha$. Moreover, all its derivatives of order at most $k$ vanish on the critical set. The p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.11586","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/2608.11586/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"}