  The main aim of this paper is to extend one of the main results of Iwaniec
and Onninen (Arch. Ration. Mech. Anal., 194: 927-986, 2009). We prove that, the
so called total energy functional defined on the class of radial streachings
between annuli attains its minimum on a total energy diffeomorphism between
annuli. This involves a subtle analysis of some special ODE.
