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.
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