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

1. 设Ω为n维欧几里得空间中的一个域，并考虑初始数据为正常数的初始-狄利克雷问题。设D是一个满足内锥条件的定义域，且D^^- Ω。我们考虑了边界∂D是解的平稳等温面的问题，得到了以下两个定理：(i)设Ω为有界域或满足外球面条件的外域。如果∂D是一个静止的等温曲面，那么∂Ω一定是一个球体。（ii）设Ω是一个满足一致外球条件的无界域，并假设∂Ω包含一个非空开子集，其中∂Ω相对于∂Ω的外法线方向的主曲率是非负的。此外，假设对于任意r >，∂Ω包含半径为r >的（N -1）维球上的图。如果∂D是一个静止的等温曲面，那么∂Ω必须是一个超平面或两个平行的超平面。这是Chamberland和Siegel（1997）关于热方程解的热点的一个猜想。设Ω为欧几里得空间中包含原点的有界域，并考虑初始数据为正常数的初始-狄利克雷问题。该猜想表明，如果原点是一个平稳热点，那么Ω在正交变换的一个本质子群G的作用下是不变的。关于这个猜想，当空间维度为二时，我们得到了以下四个定理：(i)设Ω为三角形。如果原点是一个静止的热点，那么Ω必须是一个以原点为中心的等边三角形。（ii）设Ω为凸四边形，则Ω必须为以原点为圆心的平行四边形。（iii）若原点为静止热点，则Ω不是非凸四边形。（iv）设Ω为凸m多边形（m = 5或6）。假设以原点为圆心的内切圆与Ω的每条边都有接触，并假设原点是一个静止的热点。然后，如果m = 5， Ω一定是一个以原点为中心的正五边形，如果m = 6， Ω在三个角（π/3、2π/3和π）之一的旋转下一定是不变的。少

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