The paper establishes the existence of homeomorphisms between two planar domains that minimize the Dirichlet energy.Among all homeomorphisms f : Ω onto −→ Ω * between bounded doubly connected domains such that Mod ΩMod Ω * there exists, unique up to conformal authomorphisms of Ω, an energy-minimal diffeomorphism.No boundary conditions are imposed on f . Although any energyminimal 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.2000 Mathematics Subject Classification. Primary 58E20; Secondary 30C62, 31A05.

Abstract. Every homeomorphism h : X → Y between planar open sets that belongs to the Sobolev class W 1,p (X, Y), 1 < p < ∞, can be approximated in the Sobolev norm by C ∞ -smooth diffeomorphisms.

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We prove that planar homeomorphisms can be approximated by diffeomorphisms in the Sobolev space W 1,2 and in the Royden algebra. As an application, we show that every discrete and open planar mapping with a holomorphic Hopf differential is harmonic. onto −→ Ω * such that