{"paper":{"title":"Some isoperimetric inequalities with respect to monomial weights","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Angelo Alvino, Anna Mercaldo, Francesco Chiacchio, Friedemann Brock, Maria Rosaria Posteraro","submitted_at":"2019-07-08T14:54:50Z","abstract_excerpt":"We solve a class of isoperimetric problems on $\\mathbb{R}^2_+ :=\\left\\{ (x,y)\\in \\mathbb{R} ^2 : y>0 \\right\\}$ with respect to monomial weights. Let $\\alpha $ and $\\beta $ be real numbers such that $0\\le \\alpha <\\beta+1$, $\\beta\\le 2 \\alpha$. We show that, among all smooth sets $\\Omega$ in $\\mathbb{R} ^2_+$ with fixed weighted measure $\\iint_{\\Omega } y^{\\beta} dxdy$, the weighted perimeter $\\int_{\\partial \\Omega } y^\\alpha \\, ds$ achieves its minimum for a smooth set which is symmetric w.r.t. to the $y$--axis, and is explicitly given. Our results also imply an estimate of a weighted Cheeger c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.03659","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"}