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

• 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 /

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（空间临界点的行为和水平表面的行为）与域形状之间的关系。我们获得以下内容：1。考虑欧几里得空间中的热方程的初始值问题，初始数据是域ω的特征函数。我们引入了几何条件，即ω在γ中均匀致密，这对于溶液具有固定的等温表面γ是必不可少的。统一密集的域被分类。 （Trans。Amer。Math。Soc。，358（2006），4821-4841）2。考虑欧几里得空间中有界域ω中线性和非线性扩散方程的初始界值问题，其初始数据为零，并且具有正常恒定边界值。让B仅在一个点以ω触摸∂ω中的一个球。然后，在初始时间内，溶液积分的不对称公式涉及在……更多e点处的主曲线。这个事实解释了扩散与ω几何形状之间的关系。 （Proc。RoyalSoc。EdinburghSect。A，137（2007），373-388）3。考虑在域中的域ω中具有正常恒定初始数据的热方程的初始 - 迪里希LET问题，在欧几里得空间中具有无界边界∂Ω。在各种在ω上的全局假设下，我们证明，如果溶液具有固定的等温表面，则∂Ω由超平面组成。 （印第安纳大学数学J.，出现）4。考虑在平面中有界凸多边形域ω上具有正恒定初始数据的热方程式的初始二里比特问题。当ω具有m（m≦5）侧面，并且∂Ω的每一侧都触及铭刻的圆时，我们获得了具有固定热点的新必要条件。 （提交出版物）5。在具有零初始数据和正常恒定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，解决方案的初始行为是通过距离函数到边界的。我们将此结果扩展到借助粘度溶液理论的某些非线性扩散方程，这些非线性扩散方程是均匀的抛物线。此外，在非线性扩散方程的情况下，我们通过具有固定水平表面的溶液给出球体的表征。 （准备）少

项目成果

Stationary isothermic surfaces and uniformly dense domains

静止等温面和均匀致密域

• 发表时间: 2006
• 期刊: Transactions of the American Mathematical Society 358・11
• 作者: R.Magnanini;J.Prajapat;S.Sakaguchi
• 通讯作者: S.Sakaguchi

R.Magnanini, S.Sakaguchi: "On stationary hot spots and isothermic surfaces"Progress in Analysis, Proceedings of the 3rd International ISAAC Congress, (World Scientific). VOL.II. 877-881 (2003)

R.Magnanini、S.Sakaguchi：“关于固定热点和等温面”分析进展，第三届国际 ISAAC 大会论文集，（世界科学）。

Stationary isothermic surfaces and uniformly dense domain

静止等温面和均匀致密域

• 发表时间: 2006
• 期刊: Trans.Amer.Math.Soc. 358-11
• 作者: R.Magnanini;J.Prajapat;S.Sakaguchi
• 通讯作者: S.Sakaguchi

Interaction between degenerate diffusion and shape of domain

简并扩散与域形状之间的相互作用

• 发表时间: 2007
• 期刊: Proceedings of the Royal Society of Edinburgh, Section A 137・2
• 作者: R.Magnanini;S.Sakaguchi
• 通讯作者: S.Sakaguchi

Stationary isothermic surfaces for unbounded domains

无界域的固定等温面

• 期刊: Indiana University Mathematics Journal (to appear)
• 作者: R.Magnanini;S.Sakaguchi
• 通讯作者: S.Sakaguchi