Properties and relations for two classes of distance boundary condition domains (Q2702907)

Let \(D\) be a proper subdomain in the \(\mathbb{R}^n\). For any two points \(x_1,x_2\in D\), the quasihyperbolic distance between them is \(K_D(x_1,x_2)=\inf\limits_\gamma\int_\gamma d(x,\partial D)^{-1} ds,\) where \(\gamma\) is a rectfiable curve joining \(x_1\) and \(x_2\) and lying in \(D\). Another distance between these two points is NEWLINE\[NEWLINEj_D(x_1,x_2)={1\over 2} \log\Big(1+\frac{|x_1-x_2|}{d(x_1,\partial D}\Big)\Big(1+\frac{|x_1-x_2|}{d(x_2,\partial D}\Big). NEWLINE\]NEWLINE Let \(\rho\) be a metric in \(D\). If there exit \(x_0\in D\) and two real numbers \(a\) and \(b\) such that \(\rho_D(x,x_0)\leq a \log(b/d(x,\partial D))\), we say that \(D\) satisfies a \(\rho\) distance boundary condition and write \(D\in\rho-BC\). The main results in this note are as follows. 1) Let \(f: D\to D'\) be K-quasiconformal and \(f\in\text{ Loclip}_\beta(D), 0<\beta\leq 1.\) If \(D\in K-BC,\) then \(D'\in K-BC\) and if \(D\in j-BC\) and \(C(f,\partial D)=\partial D',\) then \(D'\in j-BC\). 2) If \(D\in K-BC\), then \(D\in j-BC\). If \(D\in j-BC\) and \(D\) is a uniform domain, then \(D\in K-BC\).