论文标题
热方程解决方案和可及集合的分析性能
Analytic properties of heat equation solutions and reachable sets
论文作者
论文摘要
最近，在该间隔内部满足热时间的时间间隔，对功能的空间有一定的兴趣。这种功能的特征是在原始间隔作为对角线的正方形上进行分析。在此简短的说明中，我们提供了一个直接的论点，即该结果的类似物符合任何维度。对于有界Lipschitz域上的热方程式，在正时，在正时，所有解决方案都可以扩展到几何确定的子域$ \ MATHCAL {E}（E}（ω）$ of $ \ MATHBB {C}^D $含有$ω$。从某种意义上说，这个域是锋利的，没有更大的域是正确的。如果$ω$是一个球，我们就会证明该定理几乎是相反的。在$ \ MATHCAL {E}（ω）$的开放社区中进行分析的任何函数都是可以在积极时间从热方程的解决方案获得的。这是基于对复合域中热方程溶液的收敛性的分析，该分析使用了热方程的边界层电位方法。相反定理是使用灯芯旋转到我们结果证明的复杂域获得的。这为文献中问题的一维分析中出现的形状提供了一个简单的解释。在这种情况下，它还提供了新的简短和概念上的证明。
There recently has been some interest in the space of functions on an interval satisfying the heat equation for positive time in the interior of this interval. Such functions were characterised as being analytic on a square with the original interval as its diagonal. In this short note we provide a direct argument that the analogue of this result holds in any dimension. For the heat equation on a bounded Lipschitz domain $Ω\subset \mathbb{R}^d$ at positive time all solutions are analytically extendable to a geometrically determined subdomain $\mathcal{E}(Ω)$ of $\mathbb{C}^d$ containing $Ω$. This domain is sharp in the sense that there is no larger domain for which this is true. If $Ω$ is a ball we prove an almost converse of this theorem. Any function that is analytic in an open neighborhood of $\mathcal{E}(Ω)$ is reachable in the sense that it can be obtained from a solution of the heat equation at positive time. This is based on an analysis of the convergence of heat equation solutions in the complex domain using the boundary layer potential method for the heat equation. The converse theorem is obtained using a Wick rotation into the complex domain that is justified by our results. This gives a simple explanation for the shapes appearing in the one-dimensional analysis of the problem in the literature. It also provides a new short and conceptual proof in that case.
