Dirichlet problem for weakly harmonic maps with rough data

Weakly harmonic maps from a domain Omega subset of R-d, d >= 2 (the upper half-space R-+(d) or a bounded C-1,C-alpha domain, alpha is an element of (0,1]) into a smooth closed manifold are studied. Prescribing small Dirichlet data in either of the classes L-infinity(partial derivative Omega) or BMO(partial derivative 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-loc(infinity). 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.