论文标题
在广义的aviles-giga功能上：紧凑，零能量状态，规律性估计和能量边界
On a generalized Aviles-Giga functional: compactness, zero-energy states, regularity estimates and energy bounds
论文作者
论文摘要
给定$ \ mathbb {r}^2 $ on $ \ c^1 $ in $ \ mathbb {r}^2 \ setMinus \ {0 \} $，我们研究了任何严格的凸出范围$ \ | \ cdot \ | $ m \ right |^2 + \ \ frac {1}ε\ left（1- \ | m \ |^2 \ right）^2 \ right）^2 \ right）\，dx，\]，dx，\]使用，如在Euclidean Case $ \ | \ cdot \ | = | \ cdot | $中，限制方程的熵概念$ \ | m \ | = 1 $，$ \ nabla \ nabla \ cdot m = 0 $，我们获得以下内容。首先，我们证明了有界能量序列的$ l^p $。其次，我们证明了零能量状态（消失的能量序列的限制），从而概括并简化了Bochard和Pegon的结果。第三，就其熵生产而言，我们获得了有界能量序列序列的最佳规律性估计。第四，对于$ bv $的限制映射，我们表明，由熵制作提供的下限和一维过渡配置文件提供的上限为相同的顺序。前两个点类似于欧几里得案中已知的$ \ | \ cdot \ | = | \ cdot | $，最后两个点对norm $ $ \ | \ cdot \ | $的各向异性敏感。
Given any strictly convex norm $\|\cdot\|$ on $\mathbb{R}^2$ that is $C^1$ in $\mathbb{R}^2\setminus\{0\}$, we study the generalized Aviles-Giga functional \[I_ε(m):=\int_Ω \left(ε\left|\nabla m\right|^2 + \frac{1}ε\left(1-\|m\|^2\right)^2\right) \, dx,\] for $Ω\subset\mathbb R^2$ and $m\colonΩ\to\mathbb R^2$ satisfying $\nabla\cdot m=0$. Using, as in the euclidean case $\|\cdot\|=|\cdot|$, the concept of entropies for the limit equation $\|m\|=1$, $\nabla\cdot m=0$, we obtain the following. First, we prove compactness in $L^p$ of sequences of bounded energy. Second, we prove rigidity of zero-energy states (limits of sequences of vanishing energy), generalizing and simplifying a result by Bochard and Pegon. Third, we obtain optimal regularity estimates for limits of sequences of bounded energy, in terms of their entropy productions. Fourth, in the case of a limit map in $BV$, we show that lower bound provided by entropy productions and upper bound provided by one-dimensional transition profiles are of the same order. The first two points are analogous to what is known in the euclidean case $\|\cdot\|=|\cdot|$, and the last two points are sensitive to the anisotropy of the norm $\|\cdot\|$.
