Estimation of Sobolev-type embedding constant on domains with minimally smooth boundary using extension operator
Kazuaki Tanaka, Kouta Sekine, Makoto Mizuguchi, Shin’ichi Oishi · Journal of Inequalities and Applications · 2015
In this paper, we propose a method for estimating the Sobolev-type embedding constant from $W^{1,q}(\Omega)$ to $L^{p}(\Omega)$ on a domain $\Omega\subset\mathbb{R}^{n}$ ( $n=2,3,\dots$ ) with minimally smooth boundary (also known as a Lipschitz domain), where $p\in(n/(n-1),\infty)$ and $q=np/(n+p)$ . We estimate the embedding constant by constructing an extension operator from $W^{1,q}(\Omega)$ to $W^{1,q}(\mathbb{R}^{n})$ and computing its operator norm. We also present some examples of estimating the embedding constant for certain domains.