球上低Aviles Giga能量函数的正则性刻划的简单证明
A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity
摘要：
Aviles Giga泛函是一个众所周知的二阶泛函，它形成了一个模型，用于起泡，并在一定的制度液晶，相关的功能模型薄磁化膜。给定Lipschitz域$Ω\subset R^2$，泛函是$I_ε(u)=1/2\int_Ω ε^{-1}|1-|Du|^2|^2+ε|D^2 u|^2$，其中$u$属于$W^{2,2}_{0}(Ω)$中的函数子集，其梯度（在迹的意义上）满足$Du(x)\cdot η_x=1$，其中$η_x$是在$x$处垂直于$\partial Ω$的向内指向单元。在Jabin，奥托，Perthame特点的一类功能，其中包括所有的限制序列$u_n\in W^{2,2}_0(Ω)$与$I_{ε_n}(u_n)\to 0$作为$ε_n\to 0$。一个推论他们的工作是，如果存在这样一个序列$(u_n)$有界域$Ω$，那么$Ω$必须是一个球和（直到改变符号）$u:=\lim_{n\to \infty} u_n =\mathrm{dist}(\cdot,\partialΩ)$。最近，我们提供了一个定量的推广，这个推论的凸域空间使用“补偿紧”启发的计算源于证明$I_ε$的davivity的DeSimone，穆勒，科恩，奥托。本文利用正则性理论和常微分方程的方法，对不需要$Du$的迹条件的情况下，给出了$Ω=B_1(0)$的更精确的估计和更简单的证明.
英文摘要：
The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain $Ω\subset R^2$ the functional is $I_ε(u)=1/2\int_Ω ε^{-1}|1-|Du|^2|^2+ε|D^2 u|^2$ where $u$ belongs to the subset of functions in $W^{2,2}_{0}(Ω)$ whose gradient (in the sense of trace) satisfies $Du(x)\cdot η_x=1$ where $η_x$ is the inward pointing unit normal to $\partial Ω$ at $x$. In Jabin, Otto, Perthame characterized a class of functions which includes all limits of sequences $u_n\in W^{2,2}_0(Ω)$ with $I_{ε_n}(u_n)\to 0$ as $ε_n\to 0$. A corollary to their work is that if there exists such a sequence $(u_n)$ for a bounded domain $Ω$, then $Ω$ must be a ball and (up to change of sign) $u:=\lim_{n\to \infty} u_n =\mathrm{dist}(\cdot,\partialΩ)$. Recently we provided a quantitative generalization of this corollary over the space of convex domains using `compensated compactness' inspired calculations originating from the proof of coercivity of $I_ε$ by DeSimone, Muller, Kohn, Otto. In this note we use methods of regularity theory and ODE to provide a sharper estimate and a much simpler proof for the case where $Ω=B_1(0)$ without the requiring the trace condition on $Du$.
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