Let Ω ⊂ R 2 be a bounded simply connected domain. We show that, for a fixed (every) p ∈ (1, ∞), the divergence equationif and only if Ω is a John domain, if and only if the weighted Poincaré inequality Ωholds for some (every) q ∈ [1, ∞). In higher dimensions similar results are proved under some additional assumptions on the domain in question.
Abstract:Let Ω ⊂ R n , n ≥ 2, be a bounded domain satisfying the separation property. We show that the following conditions are equivalent: (i) Ω is a John domain; (ii) for a fixed p ∈ (1, ∞), the Korn inequality holds for each u ∈ W 1,p (Ω, R n )satisfying Ω ∂u i ∂x j
We prove that for mappings in W 1,n (B n , Ê m ), continuous up to the boundary, with modulus of continuity satisfying a certain divergence condition, the image of the boundary of the unit ball has zero n-Hausdorff measure. For Hölder continuous mappings we also prove an essentially sharp generalized Hausdorff dimension estimate. 0 2010 Mathematics Subject Classification: 46E35, 26B35, 26B10
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.