The “hot spots” conjecture for domains with two axes of symmetry
The “hot spots” conjecture for domains with two axes of symmetry
复制标题
具有两个对称轴的域的“热点”猜想
DOI：
10.1090/s0894-0347-00-00346-5
复制
发表时间：
2000
影响因子：
3.9
通讯作者：
N. Nadirashvili
中科院分区：
文献类型：
--
作者：
D. Jerison;N. Nadirashvili
考虑一个具有两个对称轴的凸平面区域。证明了具有最低非零本征值的Neumann特征函数的最大值和最小值仅出现在边界上的点上。我们用下面的形式推导出J·劳赫的“热点”猜想。如果初始温度分布不与第一个非零特征空间正交，则温度达到最大值的点趋向于边界。实际上，如果边界具有正曲率，则最大值点在有限时间内到达边界。Bafiuelos和Burdzy[BB]已经用热方程和将初始条件变形为特征函数的概率方法证明了这类结果。这里我们介绍了一种基于区域变形的新技术。我们方法的一个优点是它即使在多个本征值的情况下也能工作。在得到结果的过程中，我们证明了对称区域上Neumann特征函数的单调性，并且与独立感兴趣的Dirichlet问题进行了尖锐的比较。
Consider a convex planar domain with two axes of symmetry. We show that the maximum and minimum of a Neumann eigenfunction with lowest nonzero eigenvalue occur at points on the boundary only. We deduce J. Rauch's "hot spots" conjecture in the following form. If the initial temperature distribution is not orthogonal to the first nonzero eigenspace, then the point at which the temperature achieves its maximum tends to the boundary. In fact the maximum point reaches the boundary in finite time if the boundary has positive curvature. Results of this type have already been proved by Bafiuelos and Burdzy [BB] using the heat equation and probabilistic methods to deform initial conditions to eigenfunctions. We introduce here a new technique based on deformation of the domain. An advantage of our method is that it works even in the case of multiple eigenvalues. On the way toward our results, we prove monotonicity properties for Neumann eigenfunctions for symmetric domains that need not be convex and deduce a sharp comparison of eigenvalues with the Dirichlet problem of independent interest.