  The paper establishes the existence of homeomorphisms between two planar
domains that minimize the Dirichlet energy. Specifically, among all
homeomorphisms f : R -> R* between bounded doubly connected domains such that
Mod (R) < Mod (R*) there exists, unique up to conformal authomorphisms of R, an
energy-minimal diffeomorphism. 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.
