{"paper":{"title":"Attainability of the fractional Hardy constant with nonlocal mixed boundary conditions. Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Abdelrazek Dieb, Ahmed Attar, Boumediene Abdellaoui, Ireneo Peral","submitted_at":"2017-09-25T09:36:47Z","abstract_excerpt":"The first goal of this paper is to study necessary and sufficient conditions to obtain the attainability of the \\textit{fractional Hardy inequality } $$\\Lambda_{N}\\equiv\\Lambda_{N}(\\Omega):=\\inf_{\\{\\phi\\in \\mathbb{E}^s(\\Omega, D), \\phi\\neq 0\\}} \\dfrac{\\frac{a_{d,s}}{2} \\displaystyle\\int_{\\mathbb{R}^d} \\int_{\\mathbb{R}^d} \\dfrac{|\\phi(x)-\\phi(y)|^2}{|x-y|^{d+2s}}dx dy} {\\displaystyle\\int_\\Omega \\frac{\\phi^2}{|x|^{2s}}\\,dx}, $$ where $\\Omega$ is a bounded domain of $\\mathbb{R}^d$, $0<s<1$, $D\\subset \\mathbb{R}^d\\setminus \\Omega$ a nonempty open set and $$\\mathbb{E}^{s}(\\Omega,D)=\\left\\{ u \\in H^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.08399","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"}