arXiv 1504.008452015-04-03math.AP

具有指定次数的Ginzburg-Landau能量极小元的存在性和不存在性

Existence and non existence results for minimizers of the Ginzburg-Landau energy with prescribed degrees

Mickaël Dos Santos, Rémy Rodiac

备注：: 31 pages

摘要：

设$\mathcal{D} =Ω\setminus\barω \subset \mathbb{R}^2$是光滑环型区域.我们考虑简化的Ginzburg-Landau能量$E_ε(u)=\frac{1}{2}\int_{\mathcal{D}} |\nabla u|^2 +\frac{1}{4ε^2}\int_{\mathcal{D}} (1-|u|^2)^2$，其中$u: \mathcal{D} \rightarrow \mathbb{C}$，并在域的边界上寻找具有指定度$deg(u,\partial Ω)=p$，$deg(u,\partial ω)=q$的$E_ε$的极小元。对于大型$ε$和平衡度，即，$p=q$，我们得到了{ \it thin}域极小解的存在性.我们还证明了不存在的极小$E_ε$，大$ε$，$p\neq q$，$pq>0$和$\mathcal{D}$的情况下，$E_ε$是一个具有大容量的圆环（对应于“薄”环）。我们的方法依赖于类似的结果获得的狄利克雷能源$E_\infty(u)=\frac{1}{2}\int_{\mathcal{D}}|\nabla u|^2$，存在的结果Berlyand和Golovaty和Misiats开发的技术。

英文摘要：

Let $\mathcal{D} =Ω\setminus\barω \subset \mathbb{R}^2$ be a smooth annular type domain. We consider the simplified Ginzburg-Landau energy $E_ε(u)=\frac{1}{2}\int_{\mathcal{D}} |\nabla u|^2 +\frac{1}{4ε^2}\int_{\mathcal{D}} (1-|u|^2)^2$, where $u: \mathcal{D} \rightarrow \mathbb{C}$, and look for minimizers of $E_ε$ with prescribed degrees $deg(u,\partial Ω)=p$, $deg(u,\partial ω)=q$ on the boundaries of the domain. For large $ε$ and for balanced degrees, i.e., $p=q$, we obtain existence of minimizers for {\it thin} domain. We also prove non-existence of minimizers of $E_ε$, for large $ε$, in the case $p\neq q$, $pq>0$ and $\mathcal{D}$ is a circular annulus with large capacity (corresponding to "thin" annulus). Our approach relies on similar results obtained for the Dirichlet energy $E_\infty(u)=\frac{1}{2}\int_{\mathcal{D}}|\nabla u|^2$, the existence result obtained by Berlyand and Golovaty and on a technique developed by Misiats.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文