{"paper":{"title":"Some isoperimetric inequalities on $\\mathbb{R} ^N$ with respect to weights $|x|^\\alpha $","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"A. Alvino, A. Mercaldo, F. Brock, F. Chiacchio, M.R. Posteraro","submitted_at":"2016-06-07T16:09:10Z","abstract_excerpt":"We solve a class of isoperimetric problems on $\\mathbb{R}^N $ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\\in [0,1]$, then among all smooth sets $\\Omega$ in $\\mathbb{R} ^N$ with fixed Lebesgue measure, $\\int_{\\partial \\Omega } |x|^k \\, \\mathscr{H}_{N-1} (dx)$ achieves its minimum for a ball centered at the origin. Our results also imply a weighted Polya-Sz\\\"ego principle. In turn, we establish radiality of optimizers in some Caffarelli-Kohn-Nirenberg inequalities, and we obtain sharp bounds for eigenvalues of some nonlinear problems."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02195","kind":"arxiv","version":2},"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"}