论文标题
为什么通常“尽可能对称”解决过度确定问题的解决方案？
Why are the solutions to overdetermined problems usually "as symmetric as possible"?
论文作者
论文摘要
在本文中，我们使用隐式函数定理研究了非等级临界函数临界点的对称性。我们表明，如果形状功能相对于某些连续的旋转组不变，则其非排定临界点（具有足够平稳边界的有界开放集）共享相同的对称性。我们还考虑了形状函数表现出平移不变性外的情况，除了仅旋转不变性。最后，我们研究了该结果对锯齿蛋白型的一个/两相过度确定问题的理论的应用。 En passant，我们简单地证明了球是在体积限制下与扭转刚度有关的Lagrangian与最大化问题相关的Lagrangian的唯一非排定关键点。我们指出的是，证明不依赖移动平面或重排技术的方法。
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.
