Dirichlet problem for weakly harmonic maps with rough data
Gael Diebou Yomgne, Herbert Koch · Communications in Partial Differential Equations · 2022
Weakly harmonic maps from a domain Ω⊂Rd, d≥2 (the upper half-space R+d or a bounded C1,α domain, α∈(0,1]) into a smooth closed manifold are studied. Prescribing small Dirichlet data in either of the classes L∞(∂Ω) or BMO(∂Ω), 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, Cloc∞. 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.