Energy minimizing N -covering maps in two dimensions
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Calculus of Variations
Energy minimizing N-covering maps in two dimensions Jonathan J. Bevan1
· Jonathan H. B. Deane1
Received: 28 July 2020 / Accepted: 27 September 2020 © The Author(s) 2020
Abstract We show√that the N -covering map, which in complex coordinates is given by u N (z) := z → z N / N |z|N −1 and where N is a natural number, is a global minimizer of the Dirichlet energy D(v) = B |∇v(x)|2 d x with respect to so-called inner and outer variations. An inner variation of u N is a map of the form u N ◦ ϕ, where ϕ belongs to the class A(B) := {ϕ ∈ H 1 (B; R2 ) : det ∇ϕ = 1 a.e., ϕ|∂ B (x) = x} and B denotes the unit ball in R2 , while an√outer variation of u N is a map of the form φ ◦ u N , 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 u N ◦ ϕ in terms of the functional I (ψ; N ) := B N |ψ R |2 + N1 |ψτ |2 dy, where ψ is a suitably defined inverse of ϕ, and ψ R 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 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 ◦ u 2 ◦ 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 D(v(·, δ)), which quantity turns out to be non-negative in general and zero only when D(v(·, δ)) = D(u 2 ). Mathematics Subject Classification 49J99 · 74B20
1 Introduction Let B be the unit ball in R2 , let N be a natural number, and, for any map u in the Sobolev space H 1 (B, R2 ), let D(u) := |∇u(x)|2 d x (1.1) B
Communicated by J. M. Ball.
B 1
Jonathan J. Bevan [email protected] Department of Mathematics, University of Surrey, Guildford GU2 7XH, United Kingdom 0123456789().: V,-vol
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be the Dirichlet energy of u. Ball showed in [1] that there is a minimizer of D(·) among functions in the class Y := {y ∈ H 1 (B; R2 ) : det ∇ y = 1 a.e. in B, y|∂ B = u N }.
(1.2)
Here, u N is the N -covering map given by 1 u N (R cos α, R sin α) := √ (R cos N α, R sin N α), N √ where 0 ≤ R ≤ 1 and 0 ≤ α < 2π. The prefactor 1/ N ensures that u N satisfies det ∇u N = 1 a.e., which, together with the observation that ∇u N is essentially bounded, implies that u N belongs to Y . This paper examines and finds evidence in support of the conjecture that u N is itself a singular (i.e. non-smooth), global minimizer of D in Y by proving that u N is the unique global m
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