Dirichlet problem for weakly harmonic maps with rough data

Abstract

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, $C_{loc}^{\infty }$. 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.