On the boundary value problem for harmonic maps of the Poincaré disc

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The boundary value problem for harmonic maps of the Poincare disc is discussed. The emphasis is on the non-smoothness of the given boundary values in the problem. Let T* be a subspace of the universal Teichmuller space, defined as a set of normalized quasisymmetric homeomorphismsh of the unit circle S1 onto itself whereh admits a quasiconformal extension to the unit disc D with a complex dilatation μ satisfying $$\iint {_\mathbb{D} }|\mu (z)|^2 \rho (z)dxdy< \infty (z = x + iy)$$ , where ρ(z)¦dz¦2 is the Poincare metric of D. LetB * be a Banach space consisting of holomorphic quadratic differentials ϕ in D with norms $$\left\| \phi \right\|_{B_ * } = \left( {\iint {_\mathbb{D} }|\phi (z)|^2 \rho ^{ - 1} (z)dxdy} \right)^{\frac{1}{2}}< \infty $$ . It is shown that for any given quasisymmetric homeomorphismh: S 1→S1 ∈ T*, there is a unique quasiconformal harmonic map of D with respect to the Poincare metric whose boundary corresponding ish and the Hopf differential of such a harmonic map belongs to B*.