Onninen: Mappings of least Dirichlet energy and their Hopf differentials

Abstract. The paper is concerned with mappings h: X onto−− → Y be-tween planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in X) of the energy-minimal mappings is established within the class H 2(X,Y) of strong limits of homeomorphisms in the Sobolev space W 1,2(X,Y) , a result of considerable interest in the mathematical models of Nonlinear Elas-ticity. The inner variation (of the independent variable in X) leads to the Hopf differential hzhz ̄ dz ⊗ dz and its trajectories. For a pair of doubly connected domains, in which X has finite conformal modulus, we establish the following principle: A mapping h ∈H 2(X,Y) is energy-minimal if and only if its Hopf-differential is analytic in X and real along ∂X. In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of cracks in X. Nevertheless, cracks are triggered only by the points in ∂Y where Y fails to be convex. Th