Improved Poincaré inequalities with weights - Drelichman, I. (2008)
In this paper we prove that if Ω ∈ Rn is a bounded John domain, the following weighted Poincaré-type inequality holds:under(inf, a ∈ R) {norm of matrix} f (x) - a norm of matrixLq (Ω, w1) ≤ C {norm of matrix} ∇ f (x) d (x)α norm of matrixLp (Ω, w2) where f is a locally Lipschitz function on Ω, d (x) denotes the distance of x to the boundary of Ω, the weights w1, w2 satisfy certain cube conditions, and α ∈ [0, 1] depends on p, q and n. This result generalizes previously known weighted inequalities, which can also be obtained with our approach. © 2008 Elsevier Inc. All rights reserved.