Elastic flow of curves with partial free boundary

Antonia Diana
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引用次数: 0

Abstract

We consider a curve with boundary points free to move on a line in \({{{\mathbb {R}}}}^2\), which evolves by the \(L^2\)-gradient flow of the elastic energy, that is, a linear combination of the Willmore and the length functional. For this planar evolution problem, we study the short and long-time existence. Once we establish under which boundary conditions the PDE’s system is well-posed (in our case the Navier boundary conditions), employing the Solonnikov theory for linear parabolic systems in Hölder space, we show that there exists a unique flow in a maximal time interval [0, T). Then, using energy methods we prove that the maximal time is \(T= + \infty \).

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部分自由边界曲线的弹性流动
我们考虑一条边界点可在 \({{\mathbb {R}}}}^2\) 中的直线上自由移动的曲线,它通过弹性能量的 \(L^2\)- 梯度流(即威尔莫尔函数和长度函数的线性组合)演化。对于这个平面演化问题,我们研究了其短时和长时存在性。一旦我们利用霍尔德空间线性抛物线系统的索隆尼科夫理论,确定了 PDE 系统在哪些边界条件下(在我们的案例中是纳维边界条件)是好求解的,我们就能证明在最大时间区间 [0, T) 中存在唯一的流。然后,我们用能量方法证明最大时间是 \(T= + \infty \)。
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