Strebel discovered (in 1962) that Teichmüller's existence theorem fails for quasiconformal maps of the unit disk. Uniqueness fails too for points in $T(D^2)$ which do not have Teichmuller representatives. See Strebel's examples in the book by Gardiner and Lakic, "Quasiconformal Teichmüller Theory," p. 177-178.
On the other hand, the uniqueness theorem in certain sense does hold (Theorem 5, Chapter 4 of the same book). More precisely, if a quasiconformal map $f: D^2\to D^2$ is a Teichmüller map, i.e., it has Beltrami differential of the Teichmüller form $t \bar{\phi}/|\phi|$ (where $\phi$ is holomorphic), then $f$ is the unique extremal quasiconformal map in its equivalence class $[f]\in T(D^2)$. In particular, $[f]$ contains no other Teichmüller maps.
On third hand, Strebel proved in 1976 ("On the existence of extremal Teichmueller mappings") that for $[f]\in T(D^2)$ given by, say, smooth boundary values, the Teichmüller existence theorem does hold.
Misha
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