Energy minimizing N-covering maps in two dimensions

Abstract

We show that the N-covering map, which in complex coordinates is given by u_{_{scriptscriptstyle {N}}}(z):=z mapsto z^{N}/sqrt{N}|z|^{N-1} and where N is a natural number, is a global minimizer of the Dirichlet energy mathbb {D}(v)=int _B |nabla v(x)|^2 , dx with respect to so-called inner and outer variations. An inner variation of u_{_{scriptscriptstyle {N}}} is a map of the form u_{_{scriptscriptstyle {N}}}circ varphi , where varphi belongs to the class mathcal {A}(B):={varphi in H^1(B;mathbb {R}^2): det nabla varphi = 1 text {a.e.}, varphi arrowvert _{partial B}(x) = x} and B denotes the unit ball in mathbb {R}^2, while an outer variation of u_{_{scriptscriptstyle {N}}} is a map of the form phi circ u_{_{scriptscriptstyle {N}}}, where phi belongs to the class mathcal {A}(B(0,1/sqrt{N})). The novelty of our approach to inner variations is to write the Dirichlet energy of u_{_{scriptscriptstyle {N}}}circ varphi in terms of the functional I(psi ;N):= int _{B} N |psi _{_R}|^2 + frac{1}{N} |psi _tau |^2 , dy, where psi is a suitably defined inverse of varphi , and psi _{_R} and psi _{tau } are, respectively, the radial and angular weak derivatives of psi , and then to minimise I(psi ;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 mathcal {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 mathcal {A}_p. 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,circ ,u_2,circ ,g, where h and g are suitable measure-preserving maps, in which u_2 is a local minimizer of the Dirichlet energy . The proof of this fact requires a careful calculation of the second variation of mathbb {D}(v(cdot ,delta )), which quantity turns out to be non-negative in general and zero only when mathbb {D}(v(cdot ,delta ))=mathbb {D}(u_2).