{"paper":{"title":"Solutions to the $\\sigma_k$-Loewner-Nirenberg problem on annuli are locally Lipschitz and not differentiable","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Luc Nguyen, Yanyan Li","submitted_at":"2020-01-13T14:13:00Z","abstract_excerpt":"We show for $k \\geq 2$ that the locally Lipschitz viscosity solution to the $\\sigma_k$-Loewner-Nirenberg problem on a given annulus $\\{a < |x| < b\\}$ is $C^{1,\\frac{1}{k}}_{\\rm loc}$ in each of $\\{a < |x| \\leq \\sqrt{ab}\\}$ and $\\{\\sqrt{ab} \\leq |x| < b\\}$ and has a jump in radial derivative across $|x| = \\sqrt{ab}$. Furthermore, the solution is not $C^{1,\\gamma}_{\\rm loc}$ for any $\\gamma > \\frac{1}{k}$. Optimal regularity for solutions to the $\\sigma_k$-Yamabe problem on annuli with finite constant boundary values is also established."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2001.04257","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/2001.04257/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"}