On separating conformal annuli and Mori's ring domain in \(\mathbb{R}^n\).
The paper contains two theorems about capacity of some important sets in \(\mathbb{R}^n.\) The first one is the proof of the conjecture of Vuorinen, the second gives an upper bound for the capacity of Mori's ring domain in \(\mathbb{R}^n.\) The following notations and definitions are needed: \(\mathbb{R}^n, n\geq 2\) -- \(n\)-dimensional Euclidean space; \(0\) -- the origin; \(e_j, j=1,2,...,n\) -- unit vectors in \(\mathbb{R}^n.\) \(\mathbb{\overline{R}}^n=\mathbb{R}^n \cup\{\infty\}\) -- one-point compactification of \(\mathbb{R}^n;\) \( x=(x_1, x_2, ...,x_n) \in \mathbb{R}^n,\) \(| x| =(x_1^2+...+x_n^2)^{\frac{1}{2}}.\) \([x, y]=\{(1-t)x+ty: 0\leq t\leq1\},\) \(x, y \in \mathbb{R}^n.\) \(B^n(a,r)\) -- the ball in \(\mathbb{R}^n\) with center \(a\) and radius \(r;\) \(\overline{B}^n(a,r)\) -- the closure of \(B^n(a,r).\) A domain in \(\mathbb {R}^n\) with exactly two components \(E\) and \(F\) is called a ring domain \((E, F)\) and \(\text{cap}(E, F)\) is the conformal capacity of \((E, F).\) A ring domain \(D\) in \(\mathbb{\overline{R}}^n\) is called a conformal annulus if there exists a Möbius transformation \(h\) such that \(f(D)=B^n(0, t)-\overline{B}^n(0, 1),\) for some \(t>1.\) A conformal annulus \((E, F)\) separates the points \(a, b \in \mathbb{R}^n\) from the points \(c, d \in \mathbb{R}^n\) if \(a, b \in E\) and \(c, d \in F.\) Theorem 1. Let \(x \in \mathbb{R}^n\backslash[0, e_1]\) and let \((E, F)\) be a conformal annulus that separates \(0, e_1\) from \(x, \infty.\) Then \[ \text{cap}(E, F)\geq\frac{\omega_{n-1}}{(\text{log }t_0)^{n-1}}, \] where \(\omega_{n-1}\) denotes the \((n-1)\)-dimensional surface area of the \((n-1)\)-dimensional sphere \(S^n(0, 1)\) and \(t_0=| x| +| x-e_1|+\sqrt{(| x| +| x-e_1| )^2-1}.\) The next theorem gives the bound for conformal capacity \(\nu_n(\alpha)\) of Mori's ring domain in \(\mathbb{R}^n\) in terms of the conformal capacity \(\tau_n\) of Teichmüller's ring domain. The complementary components of Mori's ring domain are the ray \(\left\{te_1: t\leq0\right\}\cup\{\infty\}\) and the circular arc \[ \left\{x \in S^{n-1}(0, 1): x_3=x_4=...=x_n=0, | x+e_1| \geq2 \text{cos }{\alpha}\right\}, 0<\alpha<\frac{\pi}{2}. \] Theorem 2. For \(0<\alpha<\text{arctan }2\) \(\nu_n(\alpha)\leq 2 \tau_n\left(\frac{(2\text{cot }{\alpha}-1)^2}{4\text{cot }{\alpha}}\right).\)
scientific article; zbMATH DE number 1910259
scientific article