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. 令Ω为N维欧几里得空间中的域，并考虑初始狄利克雷问题，初始数据为正常数。假设D是满足内锥条件和D^^-⊂Ω的域。我们考虑了边界 ∂D 如何是解的平稳等温面的问题，并得到了以下两个定理： (i) 设 Ω 为有界域或满足外球条件的外域。如果 ∂D 是静止等温面，则 ∂Ω 必定是球体。 (ii) 令 Ω 为满足均匀外球条件的无界域，并假设 ∂Ω 包含非空开子集，其中 ∂Ω 相对于 ∂Ω 的外部法线方向的主曲率是非负的。此外，假设对于任何 r > 0，∂Ω 包含半径 r > 0 的 (N -1) 维球上的图。如果 ∂D 是静止等温面，则 ∂Ω 必须是一个超平面或两个平行超平面。 2．这是 Chamberland 和 Siegel (1997) 关于热方程解的热点的猜想。令 Ω 为包含原点的欧几里得空间中的有界域，并考虑初始狄利克雷问题，初始数据为正常数。该猜想指出，如果原点是静止热点，则 Ω 在正交变换的基本子群 G 的作用下保持不变。关于这个猜想，当空间维数为二时，我们得到以下四个定理： (i)设Ω为三角形。如果原点是静止热点，则 Ω 必须是以原点为中心的等边三角形。 (ii) 设 Ω 为凸四边形，则 Ω 必定是以原点为中心的平行四边形。 (iii) 如果原点是静止热点，则 Ω 不是非凸四边形。 (iv) 设 Ω 为凸 m 多边形（ m = 5 或 6 ）。假设以原点为中心的内切圆接触 Ω 的每一侧，并假设原点是静止热点。那么，如果 m = 5，则 Ω 必须是以原点为中心的正五边形，如果 m = 6，则 Ω 在 π/3、2π/3 和 π 三个角度之一的旋转下必须保持不变。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）。