Dirichlet solutions on bordered Riemann surfaces and quasiconformal mappings (Q1884391)

Given a compact bordered Riemann surface \(S\) and a continuous function \(f\) on \(\partial S\), let \(H^S_f\) denote the harmonic function on \(S\) with the boundary value \(f\). It is shown that if \(\varphi_n:S_0\to S_n\) is a sequence of quasiconformal mappings such that the maximal dilatations \(K(\varphi_n)\to 1\), then \(H^{S_n}_{f\circ \varphi^{-1}_n}\circ\varphi_n\Rightarrow H^{S_0}_f\). Another result is real analyticity of \(t\mapsto H^{S_t}_{f\circ\varphi^{-1}_t}\circ\varphi_t\) for quasiconformal homeomorphisms \(\varphi_t\) defined by Beltrami differentials \(\mu_t(z)\).