Existence of energy-minimal diffeomorphisms between doubly connected domains

The paper establishes the existence of homeomorphisms between two planar domains that minimize the Dirichlet energy. Among all homeomorphisms, between bounded doubly connected domains such that Mod Ω≤Mod Ω^*there exists, unique up to conformal authomorphisms of Ω, an energy-minimal diffeomorphism. Here Mod stands for the conformal modulus of a domain. No boundary conditions are imposed on f. Although any energy-minimal diffeomorphism is harmonic, our results underline the major difference between the existence of harmonic diffeomorphisms and the existence of the energy-minimal diffeomorphisms. The existence of globally invertible energy-minimal mappings is of primary pursuit in the mathematical models of nonlinear elasticity and is also of interest in computer graphics.