论文标题
dirichlet-type映射的能量
Dirichlet-type energy of mappings between two concentric annuli
论文作者
论文摘要
令$ \ mathbb {a} $和$ \ mathbb {a _ {*}} $分别是两个非分级球形annuli，分别是$ \ Mathbb {r}^{n} $，分别配备了euclidean metric和加权量$ | y | y | y | y |^{1-n} $。令$ \ mathcal {f}（\ Mathbb {a}，\ Mathbb {a _ {*}}）$在$ \ Mathcal {w}^{1，n-1，n-1，n-1，n-1，n-1，n-1，\ Mathbb {\ Mathbb {a}，\ Mathbb {a}，\ Mathbb {a _ _ _ _ _ _ _ _} =对于$ n = 3 $，第二作者\ cite {kalaj2018}证明了dirichlet-type Energy $ \ mathcal {e} [e} [h] = \ int _ {\ mathbb {a a}}的最小化器\ frac {\ | dh（x）\ |^{n-1}}} {| h（x）|^{n-1}} dx $是某些概括性的径向差异性，其中$ h \ in \ mathcal {f}（f}（f}（\ nathbb {\ mathb {a}，\ a}，\ m athbbbbbbbb {a _}对于$ n \ geq 4 $的情况，他猜想最小化的人也是$ \ mathbb {a} $和$ \ mathbb {a _ {*}} $之间的某些广义径向差异。 本文的主要目的是考虑这种猜想。首先，我们研究以下组合能量积分的最小值：$ \ mathbb {e} [a，b] [h] [h] = \ int _ {\ mathbb {\ mathbb {a}} \ frac {a^{2} {2} {2} ow ρ（x）|^{n-1}}} {|ρ（x）|^{n-1}} dx，$$其中$ h =ρs\ in \ mathcal {f}（\ mathbb {a}，\ mathbb {a}，\ mathbb {a _ {**}}}}}} $，$，$，$，$ρ= |获得的结果是\ cite [theorem 1.1] {kalaj2018}的概括。作为一个应用程序，我们表明上述猜想对于$ n \ geq 4 $，即能量积分$ \ MATHCAL {e} [h] $的最小化器几乎是正确的，但存在一个最小化的顺序，属于广义径向映射。
Let $\mathbb{A}$ and $\mathbb{A_{*}}$ be two non-degenerate spherical annuli in $\mathbb{R}^{n}$ equipped with the Euclidean metric and the weighted metric $|y|^{1-n}$, respectively. Let $\mathcal{F}(\mathbb{A},\mathbb{A_{*}})$ denote the class of homeomorphisms in $\mathcal{W}^{1,n-1}(\mathbb{A},\mathbb{A_{*}})$. For $n=3$, the second author \cite{kalaj2018} proved that the minimizers of the Dirichlet-type energy $\mathcal{E}[h]=\int_{\mathbb{A}} \frac{\|Dh(x)\|^{n-1}}{|h(x)|^{n-1}}dx$ are certain generalized radial diffeomorphisms, where $h\in \mathcal{F}(\mathbb{A},\mathbb{A_{*}})$. For the case $n\geq 4$, he conjectured that the minimizers are also certain generalized radial diffeomorphisms between $\mathbb{A}$ and $\mathbb{A_{*}}$. The main aim of this paper is to consider this conjecture. First, we investigate the minimality of the following combined energy integral: $$ \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.
