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The Łojasiewicz–Simon inequality for the elastic flow
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2021-02-13 , DOI: 10.1007/s00526-020-01916-0
Carlo Mantegazza , Marco Pozzetta

We define the elastic energy of smooth immersed closed curves in \({\mathbb {R}}^n\) as the sum of the length and the \(L^2\)-norm of the curvature, with respect to the length measure. We prove that the \(L^2\)-gradient flow of this energy smoothly converges asymptotically to a critical point. One of our aims was to the present the application of a Łojasiewicz–Simon inequality, which is at the core of the proof, in a quite concise and versatile way.



中文翻译:

弹性流的Łojasiewicz-Simon不等式

我们将\({\ mathbb {R}} ^ n \)中平滑沉浸封闭曲线的弹性能量定义为长度和曲率的\(L ^ 2 \)-范数的总和措施。我们证明了该能量的\(L ^ 2 \)梯度流平稳渐近地收敛到一个临界点。我们的目标之一就是以一种简洁而通用的方式,来证明Łojasiewicz-Simon不等式的当前应用,这是证明的核心。

更新日期:2021-02-15
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