Lipschitz regularity of energy-minimal mappings between doubly connected Riemann surfaces

Let M and N be doubly connected Riemann surfaces with $${\mathscr {C}}^{1,\alpha }$$ boundaries nonvanishing conformal metrics $$\sigma $$ $$\wp respectively, assume that is a smooth metric bounded Gauss curvature $${\mathcal {K}}$$ finite area. Assume {H}}^\wp (M, N)$$ the class of all {W}}^{1,2}$$ homeomorphisms between {E}}^\wp : {\mathcal N)\rightarrow {\mathbf {R}}$$ Dirichlet-energy functional, where $$\overline{{\mathcal {H}}}^\wp closure $${{\mathcal in {W}}^{1,2}(M,N)$$ . By using result Iwaniec, Kovalev Onninen Iwaniec et al. (Duke Math J 162(4):643–672, 2013) minimizer, locally Lipschitz, we prove energy functional , which not diffeomorphism general, globally Lipschitz mapping onto N. Note that, this new also for flat surfaces, i.e. planar domains furnished Euclidean metric.