Estimates of best constants for weighted poincaré inequalities on convex domains

Abstract

Let 1 ≤ q ≤ p < ∞ and let C be the class of all bounded convex domains Ω in ℝn, Following the approach in [1], we show that the best constant C in the weighted Poincaré inequality equation presented for all Ω ∈ C, all Lipschitz continuous functions f on Ω, and all weights w which are any positive power of a non-negative concave function on Ω is the same as the best constant for the corresponding one-dimensional situation, where C reduces to the class of bounded intervals. Using facts from [9], we estimate the best constant. In the case q = 1 and 1 < p < ∞, our estimate is between the best constant and twice the best constant. Furthermore, when p = q = 1 or p = q = 2, the estimate is sharp. Finally, in the case where the domains in Rn are further restricted to be parallelepipeds, we obtain a slightly different form of Poincaré's inequality which is better adapted to directional derivatives and the sidelengths of the parallelepipeds. We also show that this estimate is sharp for a fixed rectangle. © 2006 London Mathematical Society.