Neohookean deformations of annuli, existence, uniqueness and radial symmetry

Let X = x ∈ ℝ²; r < {pipe}x{pipe} < R} and Y = {y ∈ ℝ²; r_* < {pipe}y{pipe} < R_* be annuli in the plane. We consider the class F(X, Y) of all orientation preserving homeomorphisms, in the Sobolev space W^1,2(X, Y) which keep the boundary circles in the same order. This means that, and,. We study the Neohookean energy integral, where Φ ∈ C^∞(0, ∞) is positive and strictly convex. We assume in addition that the function, and its derivative extend continuously to [0, ∞), with Ψ(0) = 0. Then we prove: Theorem 1 The minimum of energy within the class F (X, Y) is attained for a radial map h(x) = H({pipe}x{pipe}) x/{pipe}x{pipe}. The minimizer is C^∞-smooth and is unique up to a rotation of the annuli. We believe that not only the result but also some novelties in the computation might gain a particular interest.