Dirichlet-type energy of mappings between two concentric annuli.

Let A and A ∗ be two non-degenerate spherical annuli in R n equipped with the Euclidean metric and the weighted metric | y | 1 - n , respectively. Let F (A , A ∗) denote the class of homeomorphisms h from A onto A ∗ in the Sobolev space W 1 , n - 1 (A , A ∗) . For n = 3 , the second author (Kalaj in Adv Calc Var, 10.1515/acv-2018-0074, 2019) proved that the minimizers of the Dirichlet-type energy E [ h ] = ∫ A ‖ D h (x) ‖ n - 1 | h (x) | n - 1 d x are certain generalized radial diffeomorphisms, where h ∈ F (A , A ∗) . For the case n ≥ 4 , he conjectured that the minimizers are also certain generalized radial diffeomorphisms between A and A ∗ . The main aim of this paper is to consider this conjecture. First, we investigate the minimality of the following combined energy integral: E [ a , b ] [ h ] = ∫ A a 2 ϱ n - 1 (x) ‖ D S (x) ‖ n - 1 + b 2 | ∇ ϱ (x) | n - 1 ϱ n - 1 (x) d x , where h = ϱ S ∈ F (A , A ∗) , ϱ = | h | and a , b > 0 . The obtained result is a generalization of [Kalaj (Adv Calc Var, 10.1515/acv-2018-0074, 2019), Theorem 1.1]. As an application, we show that the above conjecture is almost true for the case n ≥ 4 , i.e., the minimizer of the energy integral E [ h ] does not exist but there exists a minimizing sequence which belongs to the generalized radial mappings. [ABSTRACT FROM AUTHOR]