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

TL;DR

This paper develops sharper estimates for the Sobolev embedding constant on domains that can be divided into convex subdomains, improving previous bounds by leveraging domain decomposition and extension operator norms.

Contribution

It introduces new, more precise estimation methods for Sobolev embedding constants on convex-dividable domains, refining prior large-value bounds.

Findings

Derived sharper bounds for embedding constants on convex domains

Demonstrated improved estimation accuracy over previous methods

Applicable to domains partitioned into convex subdomains

Abstract

This paper is concerned with an explicit value of the embedding constant from $W^{1,q}(\Omega)$ to $L^{p}(\Omega)$ for a bounded domain $\Omega\subset\mathbb{R}^N~(N\in\mathbb{N})$, where $1\leq q\leq p\leq \infty$. To obtain this value, 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, in the article (K. Tanaka, K. Sekine, M. Mizuguchi, and S. Oishi, Estimation of Sobolev-type embedding constant on domains with minimally smooth boundary using extension operator, Journal of Inequalities and Applications, Vol. 389, pp. 1-23, 2015). This formula is also applicable to a domain that can be divided into Lipschitz domains. However, the values computed by the previous formula are very large. In this paper, we propose several sharper estimations of the embedding constant on a bounded domain that can be divided into convex domains.