Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional
Euler‐lagrange equation and regularity for flat minimizers of the Willmore functional
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威尔莫尔泛函的平坦极小值的欧拉-拉格朗日方程和正则性
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发表时间：
2011
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通讯作者：
Peter Hornung
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作者：
Peter Hornung
设\inputamssym $S\subset{\Bbb R}^2$是一个具有C∞类边界的有界域，gij = δij表示\inputamssym ${\Bbb R}^2$上的平坦度量.设u是黎曼流形（S，g）到\input amssym ${\Bbb R}^3$中的所有W2，2等距浸入的一个子类（通过在流形S的部分上规定边界条件定义）中Willmore泛函的极小元。本文导出了Euler-拉格朗日方程，并研究了其正则性.我们的主要正则性结果是极小化器u是C3远离某个奇异集<$0和C∞远离一个更大的奇异集<$0 <$0。我们得到了这些奇异集的几何特征，我们推导出的缩放u及其衍生物附近。
Let \input amssym $S\subset{\Bbb R}^2$ be a bounded domain with boundary of class C∞, and let gij = δij denote the flat metric on \input amssym ${\Bbb R}^2$. Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of ∂S) of all W2,2 isometric immersions of the Riemannian manifold (S, g) into \input amssym ${\Bbb R}^3$. In this article we derive the Euler‐Lagrange equation and study the regularity properties for such u. Our main regularity result is that minimizers u are C3 away from a certain singular set Σ and C∞ away from a larger singular set Σ ∪ Σ0. We obtain a geometric characterization of these singular sets, and we derive the scaling of u and its derivatives near Σ0.