为什么超定问题的解通常是“尽可能对称的”？
Why are the solutions to overdetermined problems usually "as symmetric as possible"?
摘要：
本文利用隐函数定理研究了形泛函非退化临界点的对称性。我们证明了，如果一个形状功能是不变的一些连续的旋转群，那么它的非退化临界点（有界开集足够光滑的边界）共享相同的对称性。我们还考虑的情况下，形状功能表现出平移不变性，除了旋转不变性。最后，我们研究了这一结果在Serrin型单/两相超定问题理论中的应用。在此基础上，我们给出了一个简单的证明，证明了球是唯一的非退化的临界点的拉格朗日相关的最大化问题的扭转刚度下的体积约束。我们注意到，证明不依赖于移动平面或重排技术的方法。
英文摘要：
In this paper, we study the symmetry properties of nondegenerate critical points of shape functionals using the implicit function theorem. We show that, if a shape functional is invariant with respect to some continuous group of rotations, then its nondegenerate critical points (bounded open sets with smooth enough boundary) share the same symmetries. We also consider the case where the shape functional exhibits translational invariance in addition to just rotational invariance. Finally, we study the applications of this result to the theory of one/two-phase overdetermined problems of Serrin-type. En passant, we give a simple proof of the fact that the ball is the only nondegenerate critical point of the Lagrangian associated to the maximization problem for the torsional rigidity under a volume constraint. We remark that the proof does not rely on either the method of moving planes or rearrangement techniques.
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