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 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 datau0; 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 rotationAdby an angle 2π/d,d ≧ 2 for planar domains and give some necessary and some sufficient conditions onu0 which relate to domains invariant underAd. We also establish some conjectures.
Stationary critical points of the heat flow in the plane / R. MAGNANINI; SAKAGUCHI S.. - In: JOURNAL D'ANALYSE MATHEMATIQUE. - ISSN 0021-7670. - STAMPA. - 88:(2002), pp. 383-396.
Stationary critical points of the heat flow in the plane
MAGNANINI, ROLANDO;
2002
Abstract
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 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 datau0; 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 rotationAdby an angle 2π/d,d ≧ 2 for planar domains and give some necessary and some sufficient conditions onu0 which relate to domains invariant underAd. We also establish some conjectures.File | Dimensione | Formato | |
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