Dirichlet-type energy of mappings between two concentric annuli

Abstract

Let $\mathbb{A}$ and ${\mathbb{A}}_{\ast }$ be two non-degenerate spherical annuli in ${\mathbb{R}}^n$ equipped with the Euclidean metric and the weighted metric $|y{|}^{1-n}$, respectively. Let $\mathcal{F}(\mathbb{A},{\mathbb{A}}_{\ast })$ denote the class of homeomorphisms in ${\mathcal{W}}^{1,n-1}(\mathbb{A},{\mathbb{A}}_{\ast })$. For $n=3$, the second author \cite{kalaj2018} proved that the minimizers of the Dirichlet-type energy $\mathcal{E}[h]={\int }_{\mathbb{A}}\frac{\Vert Dh(x){\Vert }^{n-1}}{|h(x){|}^{n-1}}dx$ are certain generalized radial diffeomorphisms, where $h\in \mathcal{F}(\mathbb{A},{\mathbb{A}}_{\ast })$. For the case $n\ge 4$, he conjectured that the minimizers are also certain generalized radial diffeomorphisms between $\mathbb{A}$ and ${\mathbb{A}}_{\ast }$. The main aim of this paper is to consider this conjecture. First, we investigate the minimality of the following combined energy integral: $$\mathbb{E}[a,b][h]={\int }_{\mathbb{A}}\frac{a^2{\rho }^{n-1}(x)\Vert DS(x){\Vert }^{n-1}+b^2|\nabla \rho (x){|}^{n-1}}{|\rho (x){|}^{n-1}}dx,$$ where $h=\rho S\in \mathcal{F}(\mathbb{A},{\mathbb{A}}_{\ast })$, $\rho =|h|$ and $a,b>0$. The obtained result is a generalization of \cite[Theorem 1.1]{kalaj2018}. As an application, we show that the above conjecture is almost true for the case $n\ge 4$, i.e., the minimizer of the energy integral $\mathcal{E}[h]$ does not exist but there exists a minimizing sequence which belongs to the generalized radial mappings.

19 pages