{"paper":{"title":"Sharp bounds on the failure of the hot spots conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.MP"],"primary_cat":"math.SP","authors_text":"Alexander W. Hsu, Jaume de Dios Pont, Mitchell A. Taylor","submitted_at":"2025-08-22T12:03:27Z","abstract_excerpt":"The hot spots ratio of a domain $\\Omega\\subset \\mathbb{R}^d$ measures the degree of failure of Rauch's hot spots conjecture on that domain. We identify the largest possible value of this ratio over all connected Lipschitz domains $\\Omega\\subset \\mathbb{R}^d$, for any dimension $d$. As $d\\to \\infty$, we show that this maximal ratio converges to $\\sqrt{e}$, which asymptotically matches the previous best known upper bound by Mariano, Panzo and Wang. For $d\\ge 2$, we show that sets extremizing the hot spots ratio do not exist, and extremizing sequences must converge to a ball at a quantitative rat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.16321","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/2508.16321/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"}