(n, rho)-harmonic mappings and energy minimal deformations between annuli

摘要

We extend the main results obtained by Iwaniec and Onninen in Memoirs of the AMS (2012). In this paper, we solve the (rho, n)-energy minimization problem for Sobolev homeomorphisms between two concentric annuli in the Euclidean space R-n. Here rho is a radial metric defined in the image annulus. The key element in the proofs is the solution to the Euler-Lagrange equation for a radial harmonic mapping. This is a new contribution on the topic related to the famous J. C. C. Nitsche conjecture on harmonic mappings between annuli on the complex plane. Namely we prove that the minimum of (rho, n)-energy of diffeomorphisms between annuli is attained by a certain (rho, n)-harmonic diffeomorphisms if and only if the original annulus can be mapped onto the image annulus by a radial (rho, n)-harmonic diffeomorphisms and the last fact is equivalent with a certain inequality for annuli which we call a generalized J. C. C. Nitsche type inequality.