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 ℝN, N ≧ 2. In [MS 1], we showed that a solution u has a stationary critical point O if and only if u satisfies a certain balance law with respect to O for any time. Furthermore, we proved necessary and sufficient conditions relating the symmetry of the domain to the initial data u0; in this way, we gave a characterization of the ball in ℝN ([MS 1]) and of centrosymmetric domains ([MS 2]). In the present paper, we consider a rotation Ad by an angle 2π/d, d ≧ 2 for planar domains and give some necessary and some sufficient conditions on u0 which relate to domains invariant under Ad. We also establish some conjectures.
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