Dirichlet problem for weakly harmonic maps with rough data

Abstract Weakly harmonic maps from a domain (the upper half-space or a bounded domain, ) into a smooth closed manifold are studied. Prescribing small Dirichlet data in either of the classes or 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, Moreover, we show that boundary data can be chosen large in the underlying topologies if Ω is smooth and bounded by perturbing strictly stable smooth harmonic maps.