同心环间映射的Dirichlet型能量
Dirichlet-type energy of mappings between two concentric annuli
摘要：
设$\mathbb{A}$和$\mathbb{A_{*}}$是$\mathbb{R}^{n}$中的两个非退化球面环，分别具有欧几里德度量和加权度量$|y|^{1-n}$.令$\mathcal{F}(\mathbb{A},\mathbb{A_{*}})$表示$\mathcal{W}^{1,n-1}(\mathbb{A},\mathbb{A_{*}})$中的同胚类。对于$n=3$，第二作者\cite{kalaj2018}证明了Dirichlet型能量$\mathcal{E}[h]=\int_{\mathbb{A}} \frac{\|Dh(x)\|^{n-1}}{|h(x)|^{n-1}}dx$的极小元是某些广义径向同态，其中$h\in \mathcal{F}(\mathbb{A},\mathbb{A_{*}})$ .对于$n\geq 4$的情形，他证明极小元也是$\mathbb{A}$和$\mathbb{A_{*}}$之间的某些广义径向同构。本文的主要目的是考虑这个猜想。首先，我们研究了以下组合能量积分的极小性：$$ \mathbb{E}[a,b][h] =\int_{\mathbb{A}}\frac{a^{2}ρ^{n-1}(x)\|DS(x)\|^{n-1}+b^{2}|\nabla ρ(x)|^{n-1}}{|ρ(x)|^{n-1}}dx, $$其中$h=ρS\in \mathcal{F}(\mathbb{A},\mathbb{A_{*}})$，$ρ=|h|$和$a,b>0$。所得结果是\cite [定理1.1]{kalaj 2018}的推广。作为应用，我们证明了上述猜想对于$n\geq 4$情形几乎成立，即，不存在能量积分的极小值$\mathcal{E}[h]$，但存在属于广义径向映射的极小值序列.
英文摘要：
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 in $\mathcal{W}^{1,n-1}(\mathbb{A},\mathbb{A_{*}})$. For $n=3$, the second author \cite{kalaj2018} proved that the minimizers of the Dirichlet-type energy $\mathcal{E}[h]=\int_{\mathbb{A}} \frac{\|Dh(x)\|^{n-1}}{|h(x)|^{n-1}}dx$ are certain generalized radial diffeomorphisms, where $h\in \mathcal{F}(\mathbb{A},\mathbb{A_{*}})$. For the case $n\geq 4$, 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: $$ \mathbb{E}[a,b][h] =\int_{\mathbb{A}}\frac{a^{2}ρ^{n-1}(x)\|DS(x)\|^{n-1}+b^{2}|\nabla ρ(x)|^{n-1}}{|ρ(x)|^{n-1}}dx, $$ where $h=ρS\in \mathcal{F}(\mathbb{A},\mathbb{A_{*}})$, $ρ=|h|$ 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 $n\geq 4$, 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.
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