Mappings of Least Dirichlet Energy and their Hopf Differentials

The paper is concerned with mappings h: X onto→ between 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 H̄₂(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 elasticity. The inner variation of the independent variable in X leads to the Hopf differential h_zh_z̄dz⊗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 ∈ H̄₂(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 slits in X (cognate with cracks). Slits are triggered by points of concavity of Y. They originate from ∂X and advance along vertical trajectories of the Hopf differential toward X where they eventually terminate, so no crosscuts are created.