The paper establishes the existence of homeomorphisms between two planar domains that minimize the Dirichlet energy. Specifically, among all homeomorphisms f : omega -> omega* between bounded doubly connected domains such that Mod (omega) < Mod (omega*) there exists, unique up to conformal authomorphisms of omega, 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.
Iwaniec, Tadeusz; Koh, Ngin-Tee; Kovalev, Leonid V.; and Onninen, Jani, "Existence of Energy-Minimal Diffeomorphisms Between Doubly Connected Domains" (2010). Mathematics Faculty Scholarship. 52.
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