{"paper":{"title":"Regularity of viscosity solutions of the $\\sigma_k$-Loewner-Nirenberg problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Jingang Xiong, Luc Nguyen, Yanyan Li","submitted_at":"2022-03-10T09:32:49Z","abstract_excerpt":"We study the regularity of the viscosity solution $u$ of the $\\sigma_k$-Loewner-Nirenberg problem on a bounded smooth domain $\\Omega \\subset \\mathbb{R}^n$ for $k \\geq 2$. It was known that $u$ is locally Lipschitz in $\\Omega$. We prove that, with $d$ being the distance function to $\\partial\\Omega$ and $\\delta > 0$ sufficiently small, $u$ is smooth in $\\{0 < d(x) < \\delta\\}$ and the first $(n-1)$ derivatives of $d^{\\frac{n-2}{2}} u$ are H\\\"older continuous in $\\{0 \\leq d(x) < \\delta\\}$. Moreover, we identify a boundary invariant which is a polynomial of the principal curvatures of $\\partial\\Ome"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2203.05254","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/2203.05254/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"}