Stability of spheres under volume-preserving mean curvature flow
Stability of spheres under volume-preserving mean curvature flow
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保体积平均曲率流下球体的稳定性
DOI：
10.4310/dpde.2010.v7.n4.a3
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发表时间：
2010
影响因子：
1.3
通讯作者：
I. Sigal
中科院分区：
文献类型：
--
作者：
D. Antonopoulou;G. Karali;I. Sigal
我们对 J. Escher 和 G. Simonett 提出的定理给出了新的基本证明，即对于接近欧几里得球体的初始条件，体积保持平均曲率流的解收敛于欧几里得球体（通常与初始球体不同）。我们的结果采用 Sobolev 规范给出的度量标准。虽然 J. Escher 和 G. Simonett 的证明广泛地使用了无限维不变流形理论和拟线性抛物型微分方程的结果，但我们的主要观点是使用欧几里得球体流形附近的解和李雅普诺夫泛函的微分不等式的正交分解。除了沿着标准线证明的局部适定性之外，我们的证明是完全独立的。
We give a new, elementary proof of the theorem, due to J. Escher and G. Simonett, that for the initial conditions close to Eucleadian spheres the solutions of the volume-preserving mean curvature flow converge to Eucleadian spheres (which, in general, differ from the initial spheres). Our result is in the metric given by Sobolev norms. While the proof by J. Escher and G. Simonett uses extensively rather involved results from the infinite-dimensional invariant manifold theory and quasilinear parabolic differential equations, our main point is to use an orthogonal decomposition of the solutions near the manifold of Euclidean spheres and differential inequalities for the Lyapunov functionals. Apart from local well-posedness, which is proven along standard lines, our proof is completely self-contained.