A compactness result in the gradient theory of phase transitions
A compactness result in the gradient theory of phase transitions
复制标题
DOI：
10.1017/s030821050000113x
复制
发表时间：
2001-01-01
影响因子：
1.3
通讯作者：
Otto, F
中科院分区：
文献类型：
--
作者：
DeSimone, A;Müller, S;Otto, F
本文研究奇摄动变分问题E-是（psi）的元素= integral是（-1）（1 - \delpsi\（2））（2）+ \delpsi\（2）的元素.当λ--> 0时，该泛函倾向于\ del psi \ = 1，并惩罚\ del psi \集中的奇点。我们的主要结果是一个紧性定理：如果{E-ψ（psi（ψ））}（向下箭头0）是一致有界的，则{del psi（ψ）}（向下箭头0）在L-2中是紧的。因此，在极限ε--> 0时，psi几乎在任何地方都能解程函方程\ del psi \ = 1。我们的分析使用“熵关系”和“div-curl引理”，采用Tartar的方法来处理线性微分方程和非线性代数关系的相互作用。
We examine the singularly perturbed variational problemE-is an element of (psi) = integral is an element of (-1) (1 - \ del psi \ (2))(2) + \ del del psi \ (2)in the plane. As epsilon --> 0, this functional favours \ del psi \ = 1 and penalizes singularities where \ del del psi \ concentrates. Our main result is a compactness theorem: if {E-epsilon(psi (epsilon))}(epsilon down arrow0) is uniformly bounded, then {del psi (epsilon)}(epsilon down arrow0) is compact in L-2. Thus, in the limit epsilon --> 0, psi solves the eikonal equation \ del psi \ = 1 almost everywhere. Our analysis uses 'entropy relations' and the 'div-curl lemma,' adopting Tartar's approach to the interaction of linear differential equations and nonlinear algebraic relations.