Stationary critical points of the heat flow in the plane

In previous papers MS 1, 2], we considered stationary critical points of solutions of the initial-boundary value problems for the heat equation on bounded domains in ℝ^N,N ≧ 2. In MS 1], we showed that a solutionu has a stationary critical pointO if and only ifu satisfies a certain balance law with respect toO for any time. Furthermore, we proved necessary and sufficient conditions relating the symmetry of the domain to the initial datau ₀; in this way, we gave a characterization of the ball in ℝ^N(MS 1]) and of centrosymmetric domains (MS 2]). In the present paper, we consider a rotationA _dby an angle 2π/d,d ≧ 2 for planar domains and give some necessary and some sufficient conditions onu ₀ which relate to domains invariant underA _d. We also establish some conjectures. This research was partially supported by a Grant-in-Aid for Scientific Research (C) (# 10640175) and (B) (# 12440042) of the Japan Society for the Promotion of Science. The first author was supported also by the Italian MURST.