{"paper":{"title":"The spectral drop problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bozhidar Velichkov, Giuseppe Buttazzo","submitted_at":"2014-06-06T10:08:56Z","abstract_excerpt":"We consider spectral optimization problems of the form $$\\min\\Big\\{\\lambda_1(\\Omega;D):\\ \\Omega\\subset D,\\ |\\Omega|=1\\Big\\},$$ where $D$ is a given subset of the Euclidean space $\\mathbb{R}^d$. Here $\\lambda_1(\\Omega;D)$ is the first eigenvalue of the Laplace operator $-\\Delta$ with Dirichlet conditions on $\\partial\\Omega\\cap D$ and Neumann or Robin conditions on $\\partial\\Omega\\cap\\partial D$. The equivalent variational formulation $$\\lambda_1(\\Omega;D)=\\min\\left\\{\\int_\\Omega|\\nabla u|^2\\,dx+k\\int_{\\partial D}u^2\\,d\\mathcal{H}^{d-1}\\ :\\ u\\in H^1(D),\\ u=0\\hbox{ on }\\partial\\Omega\\cap D,\\ \\|u\\|"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.1627","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"}