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

偏微分方程解的空间临界点和水平面的行为以及解的形状

基本信息

批准号：
15340047
负责人：
SAKAGUCHI Shigeru
金额：
$ 7.55万
依托单位：
Ehime University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15340047/
关键词：
diffusion equation nonlinear diffusion equation initial-boundary value problem initial value problem isothermic surface hot spot heat equation level surface porous medium方程式ガスコンテントヒートコンテント一様稠密領域完備極小曲面全曲率有限有限伝播性解の初期挙動 /

项目摘要

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

该项目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）与域形状之间的关系我们得到以下结论： 1．在欧几里得空间中，初始数据为域 Ω 的特征函数，我们引入 Ω 在 Γ 中均匀稠密的几何条件，这对于解具有平稳等温表面 Γ 是必要的。均匀稠密域为。 (Trans. Amer. Math. Soc., 358 (2006), 4821-4841)2. 考虑欧几里得空间中有界域 Ω 中的线性和非线性扩散方程的初始边值问题，且初始数据为零。设 B 为 Ω 中仅在一点接触 ∂ Ω 的球，则初始时刻 B 上解的积分公式涉及主函数。 ∂ Ω 在 th … More e 点的曲率这一事实解释了 Ω 的扩散与几何形状之间的关系（Proc. Royal Soc. Edinburgh Sect. A, 137 (2007), 373-388）3。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狄利克雷问题在 Ω 的各种全局假设下，我们证明如果。解具有平稳等温面，则 ∂ Ω 由超平面组成（印第安纳大学数学 J.，即将出现）4。考虑有界凸多边形域 Ω 上具有正常数初始数据的热方程的初始狄利克雷问题。当 Ω 有 m (m≤5) 条边并且 ∂ Ω 的每条边都接触内切圆时，我们得到了 Ω 具有静止热点的新必要条件（已提交发表。 )5. 在具有零初始数据和正常数边界值的线性扩散方程的初始边界值问题中，Varadhan (1967) 的结果是通过到的距离函数来描述解的初始行为。我们借助粘度解理论将此结果扩展到一些一致抛物线的非线性扩散方程，此外，我们通过非线性扩散方程中具有静止水平面的解来描述球体。。（准备中）少