Ordering of sets, hyperbolic metrics, and harmonic measures
For sets \(D_1,\dots,D_m\) with a nonempty intersection in \(\mathbb R^n\), let \(\Omega_i\) denote the intersection of all possible unions of \(i\) different sets. The system \(\{\Omega_i\}_{i=1}^m\) is called the ordering of the system \(\{D_i\}_{i=1}^m\). The ordering \(\{g_i(x)\}_{i=1}^m\) of the system of functions \(\{f_i(x)\}_{i=1}^m\), \(f_i:D_i\to R_+\), is defined as follows. Let \(x\in \Omega_i\) and \(D_{s_1},\dots, D_{s_k}\) contain \(x\). If \(s_1,\dots,s_k\) are chosen in such a way that \(f_{s_1}(x)\leq\dots\leq f_{s_k}(x)\), then by definition \(g_i(x)=f_{s_{k-m+i}}(x)\). Let \(\mu:\mathbb R\to\mathbb R_+\) be a convex function. The main result of the article is Theorem 1 according to which functions \(u_i\in C^2(D_i)\), \(v_i\in C^2(\Omega_i)\) such that \[ \Delta u_i=\mu(u_i)+f_i,\quad \Delta v_i\geq\mu(v_i)+g_i, \] \[ \limsup_{z\to\partial D_i}u_i(x)=B,\quad \limsup_{z\to\partial \Omega_i}v_i(x)\leq C, C\leq B, \] satisfy the inequality \[ \sum_{k=i}^mv_k(x)\leq\sum_{k=1}^{m+1-i}u_{s_k}(x) \] where the last summation is taken over an arbitrary choice of pairwise distinct indices \(s_1,\dots\), \(s_{m+1-i}\). The equality sign is possible only in trivial cases. Theorem 1 implies inequalities between hyperbolic metrics on \(D_i\) and \(\Omega_i\). Similar results are stated for harmonic measures of a system of compact sets and its ordering. This gives a key to solve a problem of \textit{S. Segawa} [Proc. Am. Math. Soc. 103, No. 1, 177-183 (1988; Zbl 0644.30026)] on the minimum of the harmonic measure \(\omega(0,E,U)\) among all compact subsets \(E\subset[-1,1]\) of the unit disk \(U\) having the given linear measure. Under some additional conditions a solution to Segawa's problem was found earlier by \textit{D. Betsacos} [Colloq. Math. 73, No. 2,221-228 (1997; Zbl 0874.30020)].
ordering hyperbolic metrics harmonic measure capacity
scientific article; zbMATH DE number 1364720