  Weakly harmonic maps from a domain $\Omega$ (the upper half-space $\Rd$ or a
bounded $C^{1,\alpha}$ domain, $\alpha\in (0,1]$) into a smooth closed manifold
are studied. Prescribing small Dirichlet data in either of the classes
$L^{\infty}(\partial\Omega)$ or $BMO(\partial\Omega)$, we establish solvability
of the resulting boundary value problems by means of a nonvariational method.
As a by-product, solutions are shown to be locally smooth, $C^{\infty}_{loc}$.
Moreover, we show that boundary data can be chosen large in the underlying
topologies if $\Omega$ is smooth and bounded by perturbing strictly stable
smooth harmonic maps.

Analysis of PDEs