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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.

Estimation of Sobolev-type embedding constant on domains with minimally smooth boundary using extension operator