On the existence of harmonic mappings between doubly connected domains

While the existence of conformal mappings between doubly connected domains is characterized by their conformal moduli, no such characterization is available for harmonic diffeomorphisms. Intuitively, one expects their existence if the domain is not too thick compared to the codomain. We make this intuition precise by showing that for a Dini-smooth doubly connected domain $\Omega^*$ there exists $\epsilon>0$ such that for every doubly connected domain $\Omega$ with $\operatorname{Mod} \Omega^*<\operatorname{Mod}\Omega<\operatorname{Mod} \Omega^*+\epsilon$ there exists a harmonic diffeomorphism from $\Omega$ onto $\Omega^*$.