On double-covering stationary points of a constrained Dirichlet energy

Abstract

The double-covering map udc:R2→R2 is given byudc(x)=12|x|(x22−x122x1x2) in cartesian coordinates. This paper examines the conjecture that udc is the global minimizer of the Dirichlet energy I(u)=∫B|∇u|2dx among all W1,2 mappings u of the unit ball B⊂R2 satisfying (i) u=udc on ∂B, and (ii) det∇u=1 almost everywhere. Let the class of such admissible maps be A. The chief innovation is to express I(u) in terms of an auxiliary functional G(u−udc), using which we show that udc is a stationary point of I in A, and that udc is a global minimizer of the Dirichlet energy among members of A whose Fourier decomposition can be controlled in a way made precise in the paper. By constructing variations about udc in A using ODE techniques, we also show that udc is a local minimizer among variations whose tangent ψ to A at udc obeys G(ψo)>0, where ψo is the odd part of ψ. In addition, a Lagrange multiplier corresponding to the constraint det∇u=1 is identified by an analysis which exploits the well-known Fefferman–Stein duality.