Weighted Poincaré inequalities on convex domain

Abstract

Let Ω be a bounded open convex set in Mn. Suppose that a ≥ 0, β ∈ ℝ, 1 ≤ p ≤ q < ∞, and (iquestion) Let ρ(x) = dist(x,Ωc) = min{|x - y|: y ∈ Ωc} denote the Euclidean distance to the complement of Ω. Define ρa(Ω) = fΩ ρ (x) αdx, and denote (iquestion) We derive the following weighted Poincaré inequality for locally Lipschitz continuous functions f on Ω: (iquestion) where η is the eccentricity of Ω and C is a constant depending only on p, q, α, β and the dimension n. The main point of the estimate is the way the constant depends on η for a general convex domain. We also consider the case 1 ≤ q < p 0. When q ≥ p, the case of convex domains which are symmetric with respect to a point was settled in [CD], and our estimate for q ≥ p extends that result to nonsymmetric domains. Moreover, the exponent of η is sharp and the conditions are necessary. © International Press 2010.