IKEHATA Masaru Gunma University, Faculty of Engineering, Professor, 工学部, 教授 (90202910)
HASHIMOTO Takahiro Ehime University, Faculty of Science, Instructors, 理学部, 助手 (60291499)
YANAGI Shigenori Ehime University, Faculty of Science, Associate Professor, 理学部, 助教授 (10253296)
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.
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