Breaking through borders with σ-harmonic mappings

We consider mappings U=(u1,u2), whose components solve an arbitrary elliptic equation in divergence form in dimension two, and whose respective Dirichlet data φ1,φ2 constitute the parametrization of a simple closed curve γ. We prove that, if the interior of the curve γ is not convex, then we can find a parametrization Φ=(φ1,φ2) such that the mapping U is not invertible.