{"paper":{"title":"About H\\\"older-regularity of the convex shape minimizing {\\lambda}2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Jimmy Lamboley (CEREMADE)","submitted_at":"2010-10-29T15:04:06Z","abstract_excerpt":"In this paper, we consider the well-known following shape optimization problem: $$\\lambda_2(\\Omega^*)=\\min_{\\stackrel{|\\Omega|=V_0} {\\Omega\\textrm{ convex}}} \\lambda_2(\\Omega),$$ where $\\lambda_2(\\Om)$ denotes the second eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions in $\\Om\\subset\\R^2$, and $|\\Om|$ is the area of $\\Om$. We prove, under some technical assumptions, that any optimal shape $\\Omega^*$ is $\\mathcal{C}^{1,\\frac{1}{2}}$ and is not $\\C^{1,\\alpha}$ for any $\\alpha>\\frac{1}{2}$. We also derive from our strategy some more general regularity results, in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.6239","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":""},"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"}