English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ R. Let u be a solution to Lu = div ~f in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the L(Ω, dμ) norm of |∇u| is dominated by the L(Ω, dv) norms of div ~f and |~f |. If we replace |∇u| by a local Holder norm of u, the conditions on μ and ν can be significantly weaker. Introduction. The intent of this paper is to establish sufficient conditions on two measures, μ and ν, defined on a bounded domain Ω in R, n ≥ 3, so that (1) (\ Ω |∇u(x)| dμ(x) )1/q ≤ C (\ Ω (|div ~ f(x)| + |~ f(x)|) dν(x) )1/p if 2 < p ≤ q < 2 + e, for any function u(x) that is a weak solution to { Lu(x) = div ~ f(x), x ∈ Ω, u(x)|∂Ω = 0. Here Ω is assumed to satisfy an exterior cone condition, ~ f ∈ H(Ω), and L = ∑ 1≤i,j≤n ∂ ∂xi ( ai,j(x) ∂ ∂xj ) 2000 Mathematics Subject Classification: 35J25, 42B25.

A Littlewood–Paley type inequality with applications to the elliptic Dirichlet problem