凸区域上低Aviles Giga能量函数的一个定量刻画
A quantitative characterisation of functions with low Aviles Giga energy on convex domains
摘要：
设L（U）是二阶索伯列夫空间中梯度（迹意义下）等于垂直于U的指向内单位的函数子集。L（U）上的Aviles Giga泛函作为与液晶和薄膜泡罩中的问题有关的模型，它也是Modica Mortola泛函的最自然的高阶推广。Jabin，奥托，Perthame的特点是一类功能，其中包括所有的限制序列的Aviles千兆能源为零。他们的工作的一个推论是，如果存在这样一个序列的有界域U，那么U必须是一个球和极限函数必须是距离边界。我们证明了一个定量的推广这一推论的类有界凸集。因此，我们证明了如果U有C^2边界并且靠近一个球，那么对于所有足够小的\ep，I_{\ep}的极小值接近于离边界的距离函数。
英文摘要：
Given a connected Lipschitz domain U we let L(U) be the subset of functions in 2nd order Sobolev space whose gradient (in the sense of trace) is equal to the inward pointing unit normal to U. The the Aviles Giga functional over L(U) serves as a model in connection with problems in liquid crystals and thin film blisters, it is also the most natural higher order generalisation of the Modica Mortola functional. Jabin, Otto, Perthame characterised a class of functions which includes all limits of sequences whose Aviles Giga energy goes to zero. A corollary to their work is that if there exists such a sequence for a bounded domain U, then U must be a ball and the limiting function must be the distance from the boundary. We prove a quantitative generalisation of this corollary for the class of bounded convex sets. As a consequence of this we show that if U has C^2 boundary and is close to a ball, then for all small enough \ep the minimiser of I_{\ep} is close to the distance function from the boundary.
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