论文标题
在热含量功能及其关键领域上
On the heat content functional and its critical domains
论文作者
论文摘要
我们在分析性riemannian歧管中研究和分类平滑界域，这对于始终t> 0至关重要。我们通过首先计算热量含量的第一个变化来做到这一点，然后在且仅当它具有所谓的恒定流属性时表明域至关重要，以便我们可以使用作者建立的先前的分类结果。结果是，只有当它承认等于叶子的叶子，叶子的叶子都与边界平行并且具有恒定的平均曲率时，域始终对热量含量至关重要。然后，我们考虑由退出时间矩给出的功能序列，从而概括了扭转刚度。我们证明，如果并且仅在始终对热量含量至关重要的情况下，那么一个域对于所有退出时间时刻至关重要，然后我们也会得到分类。本文的主要目的是了解普通等植物的变异特性及其在PDE理论中的作用。在某些方面，等含量叶子通过圆形球体概括了欧几里得空间的叶子的特性。
We study and classify smooth bounded domains in an analytic Riemannian manifold which are critical for the heat content at all times t>0. We do that by first computing the first variation of the heat content, and then showing that a domain is critical if and only if it has the so-called constant flow property, so that we can use a previous classification result established by the author. The outcome is that a domain is critical for the heat content at all times if and only if it admits an isoparametric foliation, that is, a foliation whose leaves are all parallel to the boundary and have constant mean curvature. Then, we consider the sequence of functionals given by the exit-time moments, which generalize the torsional rigidity. We prove that a domain is critical for all exit time moments if and only it is critical for the heat content at all times, and then we get a classification as well. The main purpose of the paper is to understand the variational properties of general isoparametric foliations and their role in PDE's theory; in some respect isoparametric foliations generalize the properties of the foliation of Euclidean space by round spheres.
