Behavior of spatial critical points and zeros of solutions of partial differential equations

偏微分方程解的空间临界点和零点的行为

• 批准号: 12440042
• 负责人: SAKAGUCHI Shigeru
• 金额: $ 4.86万
• 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: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440042/
• 关键词: heat equation; hot spot; isothermic surface; symmetry; initial-Dirichlet problem; sphere; hypersurface; polygon; 初期斉次Diricllet問題; 非有界領域; 空間臨界点; 初期斉次デリクレ問題; 双曲空間

1. Let Ω be a domain in the N-dimensional Euclidean space, and consider the initial-Dirichlet problem for initial data being a positive constant. Suppose that D is a domain satisfying the interior cone condition and D^^-⊂Ω. We considered the question how the boundary ∂D is a stationary isothermic surface of the solution, and obtained the following two theorems : (i) Let Ω be either a bounded domain or an exterior domain satisfying the exterior sphere condition. If ∂D is a stationary isothermic surface, then ∂Ω must be a sphere. (ii) Let Ω be an unbounded domain satisfying the uniform exterior sphere condition, and suppose that ∂Ω contains a nonempty open subset where the principal curvatures of ∂Ω with respect to the exterior normal direction to ∂Ω are nonnegative. Furthermore, assume that, for any r > 0, ∂Ω contains the graph over a (N -1)-dimensional ball with radius r > 0. If ∂D is a stationary isothermic surface, then ∂Ω must be either a hyperplane or two parallel hyperplanes.2. Th … More ere is a conjecture of Chamberland and Siegel (1997) concerning the hot spots of solutions of the heat equation. Let Ω be a bounded domain in the Euclidean space containing the origin, and consider the initial-Dirichlet problem for initial data being a positive constant. The conjecture stated that if the origin is a stationary hot spot, then Ω is invariant under the action of an essential subgroup G of orthogonal transformations. Concerning this conjecture, we obtained the following four theorems when the space dimension is two : (i) Let Ω be a triangle. If the origin is a stationary hot spot, then Ω must be an equilateral triangle centered at the origin. (ii) Let Ω be a convex quadrangle, then Ω must be a parallelogram centered at the origin. (iii) If the origin is a stationary hot spot, then Ω is not a non-convex quadrangle. (iv) Let Ω be a convex m-polygon ( m = 5 or 6 ). Suppose that the inscribed circle centered at the origin touches every side of Ω, and suppose that the origin is a stationary hot spot. Then, if m = 5, Ω must be a regular pentagon centered at the origin, and if m = 6, Ω must be invariant under the rotation of one of three angles, π/3, 2π/3, and π. Less

1。让ω是n维欧几里得空间中的一个域，并考虑初始数据的初始数据是一个正常数。假设d是满足内部锥体条件和d ^^ - ⊂Ω的域。我们考虑了一个问题，即边界∂D是溶液的固定等温表面，并获得了以下两个定理：（i）让ω为满足外部球体条件的有界域或外部结构域。如果∂D是固定的等温表面，则∂Ω必须是一个球体。 （ii）令ω为满足均匀外部球体条件的无界域，并假设∂Ω包含一个非空的开放子集，其中相对于外部正常方向的主曲率相对于到∂Ω的外部正常方向是非负的。此外，假设对于任何r> 0，∂Ω都包含（n -1）二维球上的图形radius r> 0。这是Chamberland和Siegel（1997）的概念，涉及热方程式的热点。令ω为包含原点的欧几里得空间中的一个有界域，并考虑初始数据的初始二核酸酯问题为正常数。该概念指出，如果原点是静止的热点，则ω在正交转换的基本亚组G的作用下是不变的。关于这个猜想，当空间维度为两个时，我们获得了以下四个定理：（i）让ω为三角形。如果原点是固定的热点，则ω必须是以原点为中心的相等三角形。 （ii）令ω为凸四边形，那么ω必须是以原点为中心的平行四边形。起源。 （iii）如果原点是固定的热点，则ω不是非凸四个四边形。 （iv）令ω为凸M-polygon（M = 5或6）。假设以原点为中心的铭文圆触摸ω的每一侧，并假设原点是固定的热点。然后，如果m = 5，则必须是以原点为中心的常规五角形，如果m = 6，则必须在三个角度之一的旋转下不变，π/3、2π/3和π必须是不变的。较少的

项目成果

S.Sakaguchi: "Stationary critical points of the heat flow in spaces of constant curvature"Journal London, Mathematical Society. 63-2. 400-412 (2001)

S.Sakaguchi：“恒定曲率空间中热流的稳态临界点”伦敦杂志，数学会。

S.Sakaguchi: "Behavior of spatial critical points and zeros of solutions of diffusion equations"Sugaku (Japanese). 54-3. 249-264 (2002)

S.Sakaguchi：“扩散方程的空间临界点和零点的行为”Sugaku（日语）。

坂口 茂: "拡散方程式の解の空間臨界点と零点の挙動"数学. 54・3. 249-264 (2002)

坂口茂：“扩散方程的空间临界点和零点的行为”54・3（2002）。

R.Magnanini and S.Sakaguchi: "Matzoh ball soup : Heat conductors with a stationary isothermic surface"Annals of Mathematics. 156-3. 931-946 (2002)

R.Magnanini 和 S.Sakaguchi：“Matzoh 球汤：具有固定等温表面的热导体”数学年鉴。

S.Sakaguchi: "Behavior of spatial critical points and zeros of solutions of diffusion equations"Sugaku Expositions (English translation). (to appear).

S.Sakaguchi：“扩散方程的空间临界点和零点的行为”Sugaku Expositions（英文翻译）。