Journal of Inequalities and Applications 
Volume 2007 (2007), Article ID 34138, 13 pages
doi:10.1155/2007/34138
Research Article

On the Precise Asymptotics of the Constant in Friedrich's Inequality for Functions Vanishing on the Part of the Boundary with Microinhomogeneous Structure

G. A. Chechkin,1 Yu. O. Koroleva,1 and L.-E. Persson2

 1Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow Lomonosov State University, Moscow 119991, Russia
2Department of Mathematics, Luleå University of Technology, Luleå 97187, Sweden
Received 3 April 2007; Revised 28 June 2007; Accepted 23 October 2007

Recommended by Michel Chipot

Abstract

We construct the asymptotics of the sharp constant in the Friedrich-type inequality for functions, which vanish on the small part of the boundary Γ1ɛ. It is assumed that Γ1ɛ consists of (1/δ)n−1 pieces with diameter of order O(ɛδ). In addition, δ=δ(ɛ) and δ→0 as ɛ→0.