热流、热含量与等参性质
Heat flow, heat content and the isoparametric property
摘要：
设$M$是一个黎曼流形，$Ω$是$M$的一个具有光滑边界的紧致域.研究了具有常单位初值条件和Dirichlet边界条件的$Ω$上热方程的解.本文的目的是研究区域的几何，对于任何固定的时间值，解（热流）的法向导数是边界上的一个常数函数。我们通过说这样的区域具有恒流性质来表达这个事实。在常曲率空间中，已知的这种域的例子是由测地线球给出的，更一般地说，是由边界连通和等参的域给出的。问题是：他们都是这样的吗？本文对这个问题给出了肯定的回答：事实上，我们更一般地证明了：如果解析黎曼流形中的区域具有常流性质，则其边界的每个分支都是等参超曲面。对于空间型，我们还涉及的顺序消失的热含量与固定的边界数据的恒定性的$r$ -中曲率的边界和等参属性。最后，我们讨论了常流性质与其他著名的超定问题，涉及拉普拉斯算子，如Serrin问题或希弗问题。
英文摘要：
Let $M$ be a Riemannian manifold and $Ω$ a compact domain of $M$ with smooth boundary. We study the solution of the heat equation on $Ω$ having constant unit initial conditions and Dirichlet boundary conditions. The purpose of this paper is to study the geometry of domains for which, at any fixed value of time, the normal derivative of the solution (heat flow) is a constant function on the boundary. We express this fact by saying that such domains have the constant flow property. In constant curvature spaces known examples of such domains are given by geodesic balls and, more generally, by domains whose boundary is connected and isoparametric. The question is: are they all like that? In this paper we give an affirmative answer to this question: in fact we prove more generally that, if a domain in an analytic Riemannian manifold has the constant flow property, then every component of its boundary is an isoparametric hypersurface. For space forms, we also relate the order of vanishing of the heat content with fixed boundary data with the constancy of the $r$-mean curvatures of the boundary and with the isoparametric property. Finally, we discuss the constant flow property in relation to other well-known overdetermined problems involving the Laplace operator, like the Serrin problem or the Schiffer problem.
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