An extremal problem of quasiconformal mappings

Abstract:

In this paper, the following problem is studied. Let $\Omega _{1}$ and $\Omega _{2}$ be two domains in the complex plane with $\Omega _{1}\cap \Omega _{2}\not =\emptyset$. Suppose that $f_{j}:\Omega _{j}\to f_{j}(\Omega _{j})$ $(j=1,2)$ are two quasiconformal mappings satisfying $f_{1}|_{\Omega _{1}\cap \Omega _{2}} =f_{2}|_{\Omega _{1}\cap \Omega _{2}}$. Let $F$ be the mapping in $\Omega _{1}\cup \Omega _{2}$ defined by $F|_{\Omega _{j}}=f_{j}$ ($j=1,2$). If both $f_{1}$ and $f_{2}$ are uniquely extremal, is $F$ always uniquely extremal? It is shown in this paper that the answer to this problem is no.