Dirichlet-type energy of mappings between two concentric annuli

Abstract

Let ${\mathbb {A}}$ and ${\mathbb {A}}_{*}$ be two non-degenerate spherical annuli in ${\mathbb {R}}^{n}$ equipped with the Euclidean metric and the weighted metric $|y|^{1-n}$, respectively. Let ${\mathcal {F}}({\mathbb {A}},{\mathbb {A}}_{*})$ denote the class of homeomorphisms h from ${\mathbb {A}}$ onto ${\mathbb {A}}_{*}$ in the Sobolev space ${\mathcal {W}}^{1,n-1}({\mathbb {A}},{\mathbb {A}}_{*})$. 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 ${\mathcal {E}}[h]=\int _{{\mathbb {A}}} \frac{\Vert Dh(x)\Vert ^{n-1}}{|h(x)|^{n-1}}dx$ are certain generalized radial diffeomorphisms, where $h\in {\mathcal {F}}({\mathbb {A}},{\mathbb {A}}_{*})$. For the case $n\ge 4$, he conjectured that the minimizers are also certain generalized radial diffeomorphisms between ${\mathbb {A}}$ and ${\mathbb {A}}_{*}$. The main aim of this paper is to consider this conjecture. First, we investigate the minimality of the following combined energy integral:

$$\begin{aligned} {\mathbb {E}}[a,b][h] =\int _{{\mathbb {A}}}\frac{a^{2}\varrho ^{n-1}(x)\Vert DS(x)\Vert ^{n-1}+b^{2}|\nabla \varrho (x)|^{n-1}}{ \varrho ^{n-1}(x)}dx, \end{aligned}$$

where $h=\varrho S\in {\mathcal {F}}({\mathbb {A}},{\mathbb {A}}_{*})$, $\varrho =|h|$ 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 $n\ge 4$, 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.