Behavior of spatial critical points and level surfaces of solutions of partial differential equations and shapes of the solutions

本课题的主要目的是研究偏微分方程解的形状（空间临界点和水平面的行为）与区域形状之间的关系。我们得到：1。考虑欧几里得空间中热方程的初值问题，初始数据是一个域的特征函数Ω，我们引入了Ω在Γ中均匀致密的几何条件，这是解具有固定等温线面的必要条件Γ。对均匀稠密域进行了分类。(反式。阿米尔。数学。Soc。生物医学工程学报，358 (2006),4821-4841考虑欧几里得空间中具有零初始数据和正常边值的有界域Ω线性和非线性扩散方程的初边值问题。设B是Ω中的一个球，只在一点上接触∂Ω。那么初始时间B上解的积分的渐近公式包含∂Ω在…More e点的主曲率。这一事实解释了扩散与Ω几何形状之间的关系。2 .中国医学杂志，2007(3),373-388。考虑欧几里德空间中具有无界边界∂Ω的域Ω中具有正常数初始数据的热方程的初始- dirichlet问题。在Ω上的各种全局假设下，我们证明了如果解有一个平稳的等温曲面，那么∂Ω由超平面组成。(印第安纳大学数学。4.答案：b。考虑平面上有界凸多边形区域Ω上初始数据为正常数的热方程的初始-狄利克雷问题。当Ω有m （m≦5）条边，∂Ω的每条边都与内切圆接触时，我们得到了Ω有固定热点的一个新的必要条件。（提交出版）对于具有零初始数据和正常边值的线性扩散方程的初始边值问题，Varadhan（1967）的结果是通过到边界的距离函数来描述解的初始行为。借助黏度解理论，我们将这一结果推广到一些均匀抛物型的非线性扩散方程。此外，对于非线性扩散方程，我们通过具有平稳水平面的解给出了球的表征。（准备中）少

The main purpose of this project is to study the relationship between shape of solutions of partial differential equations (behavior of spatial critical points and level surfaces) and shape of domains. We obtain the following:1. Consider the initial value problem for the heat equation in Euclidean space with initial data being the characteristic function of a domain Ω. We introduce the geometrical condition that Ω is uniformly dense in Γ, which is necessary for the solution to have a stationary isothermic surface Γ. The uniformly dense domains are classified. (Trans. Amer. Math. Soc., 358 (2006), 4821-4841 )2. Consider the initial-boundary value problem for linear and nonlinear diffusion equations in a bounded domain Ω in Euclidean space with zero initial data and with positive constant boundary value. Let B be a ball in Ω touching ∂ Ω only at one point. Then the asymptotic formula of the integral of the solution over B at the initial time involves the principal curvatures of ∂ Ω at th … More e point. This fact explains the relationship between diffusion and the geometry of Ω. ( Proc. Royal Soc. Edinburgh Sect. A, 137 (2007), 373-388 )3. Consider the initial-Dirichlet problem for the heat equation with positive constant initial data in a domain Ω with unbounded boundary ∂ Ω in Euclidean space. Under various global assumptions on Ω, we prove that if the solution has a stationary isothermic surface, then ∂ Ω consists of hyperplanes. ( Indiana University Math. J., to appear )4. Consider the initial-Dirichlet problem for the heat equation with positive constant initial data over a bounded convex polygonal domain Ω in the plane. When Ω has m (m≦5) sides and every side of ∂ Ω touches the inscribed circle, we obtain a new necessary condition for Ω having a stationary hot spot. (submitted for the publication )5. In the initial-boundary value problem for the linear diffusion equation with zero initial data and with positive constant boundary value, a result of Varadhan (1967) is such that the initial behavior of the solution is described through the distance function to the boundary. We extend this result to some nonlinear diffusion equations, which are uniformly parabolic, with the aid of the theory of viscosity solutions. Moreover, we give a characterization of the sphere through the solution having a stationary level surface in case of nonlinear diffusion equations. (in preparation ) Less