{"paper":{"title":"Some remarks on a shape optimization problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Francesco Della Pietra","submitted_at":"2014-03-24T09:47:26Z","abstract_excerpt":"Given $\\Omega$ bounded open set of $\\mathbb R^{n}$ and $\\alpha\\in \\mathbb R$, let us consider \\[ \\mu(\\Omega,\\alpha)=\\min_{\\substack{v\\in W_{0}^{1,2}(\\Omega)\\\\v\\not\\equiv 0}} \\frac{\\displaystyle\\int_{\\Omega} |\\nabla v|^{2}dx+\\alpha \\left|\\displaystyle\\int_{\\Omega}|v|v\\,dx \\right|}{\\displaystyle\\int_{\\Omega} |v|^{2}dx}. \\] We study some properties of $\\mu(\\Omega,\\alpha)$ and of its minimizers, and, depending on $\\alpha$, we determine the set $\\Omega_{\\alpha}$ among those of fixed measure such that $\\mu(\\Omega_{\\alpha},\\alpha)$ is the smallest possible."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.5887","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"}