Abstract
We show that the N-covering map, which in complex coordinates is given by uN(z):=z↦zN/N|z|N-1 and where N is a natural number, is a global minimizer of the Dirichlet energy D(v)=∫B|∇v(x)|2dx with respect to so-called inner and outer variations. An inner variation of uN is a map of the form uN∘φ, where φ belongs to the class A(B):={φ∈H1(B;R2):det∇φ=1a.e.,φ|∂B(x)=x} and B denotes the unit ball in R2, while an outer variation of uN is a map of the form ϕ∘uN, where ϕ belongs to the class A(B(0,1/N)). The novelty of our approach to inner variations is to write the Dirichlet energy of uN∘φ in terms of the functional I(ψ;N):=∫BN|ψR|2+1N|ψτ|2dy, where ψ is a suitably defined inverse of φ, and ψR and ψτ are, respectively, the radial and angular weak derivatives of ψ, and then to minimise I(ψ;N) by considering a series of auxiliary variational problems of isoperimetric type. This approach extends to include p-growth functionals (p>1) provided the class A(B) is suitably adapted. When 1<p<2, this adaptation is delicate and relies on the deep results of Barchiesi et al. on the space they refer to in Barchiesi et al. (Arch Ration Mech Anal 224(2):743–816, 2017) as Ap. A technique due to Sivaloganthan and Spector (Arch Ration Mech Anal 196:363–394, 2010) can be applied to outer variations. We also show that there is a large class of variations of the form v=h∘u2∘g, where h and g are suitable measure-preserving maps, in which u2 is a local minimizer of the Dirichlet energy . The proof of this fact requires a careful calculation of the second variation of D(v(·,δ)), which quantity turns out to be non-negative in general and zero only when D(v(·,δ))=D(u2).