{"paper":{"title":"Improved Moser--Trudinger inequality for functions with mean value zero in $\\mathbb R^n$ and its extremal functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Van Hoang Nguyen","submitted_at":"2017-08-09T22:37:40Z","abstract_excerpt":"Let $\\Omega$ be a bounded smooth domain in $\\mathbb R^n$, $W^{1,n}(\\Omega)$ be the Sobolev space on $\\Omega$, and $\\lambda(\\Omega) = \\inf\\{\\|\\nabla u\\|_n^n: \\int_\\Omega u dx =0, \\|u\\|_n =1\\}$ be the first nonzero Neumann eigenvalue of the $n-$Laplace operator $-\\Delta_n$ on $\\Omega$. For $0 \\leq \\alpha < \\lambda(\\Omega)$, let us define $\\|u\\|_{1,\\alpha}^n =\\|\\nabla u\\|_n^n -\\alpha \\|u\\|_n^n$. We prove, in this paper, the following improved Moser--Trudinger inequality on functions with mean value zero on $\\Omega$, \\[ \\sup_{u\\in W^{1,n}(\\Omega), \\int_\\Omega u dx =0, \\|u\\|_{1,\\alpha} =1} \\int_{\\O"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.03028","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"}