Estimation of Sobolev embedding constant on a domain dividable into bounded convex domains

Abstract

Abstract This paper is concerned with an explicit value of the embedding constant from W 1 , q ( Ω ) $W^{1,q}(\Omega)$ to L p ( Ω ) $L^{p}(\Omega)$ for a domain Ω ⊂ R N $\Omega\subset\mathbb{R}^{N}$ ( N ∈ N $N\in\mathbb{N}$ ), where 1 ≤ q ≤ p ≤ ∞ $1\leq q\leq p\leq\infty$ . We previously proposed a formula for estimating the embedding constant on bounded and unbounded Lipschitz domains by estimating the norm of Stein’s extension operator. Although this formula can be applied to a domain Ω that can be divided into a finite number of Lipschitz domains, there was room for improvement in terms of accuracy. In this paper, we report that the accuracy of the embedding constant is significantly improved by restricting Ω to a domain dividable into bounded convex domains.

Keywords

• Sobolev embedding constant
• Hardy-Littlewood-Sobolev inequality
• Young inequality