On a generalized Aviles-Giga functional: compactness, zero-energy states, regularity estimates and energy bounds
On a generalized Aviles-Giga functional: compactness, zero-energy states, regularity estimates and energy bounds
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关于广义 Aviles-Giga 泛函：紧致性、零能量态、正则性估计和能量界限
DOI：
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发表时间：
2022
影响因子：
1.9
通讯作者：
G. Peng
中科院分区：
文献类型：
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作者：
X. Lamy;A. Lorent;G. Peng
给定C1上的任意严格凸范数，我们研究了满足和的广义Aviles-Giga泛函，如在欧氏情形下，极限方程的熵的概念，得到如下结果。首先，我们证明了有界能量序列在Lp中的紧性。其次，我们证明了零能态的刚性（消失能量序列的极限），推广和简化了Bochard和Pegon的结果。第三，我们得到了最优的正则性估计的极限的有界能量序列，在他们的熵生产。第四，在BV中的极限映射的情况下，我们证明了由熵产生提供的下界和由一维过渡轮廓提供的上界是同一阶的。前两点类似于欧几里得情形中已知的情况，后两点对范数的各向异性敏感
Abstract Given any strictly convex norm on that is C 1 in we study the generalized Aviles-Giga functional for and satisfying Using, as in the euclidean case the concept of entropies for the limit equation we obtain the following. First, we prove compactness in Lp 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 and the last two points are sensitive to the anisotropy of the norm