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Stationary critical points of the heat flow in the plane

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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 datau 0; 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 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.

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Authors and Affiliations

  1. Dipartimento Di Matematica “U. Dini”, Universitá Di Firenze, Vlale Morgagni 67/A, 50134, Firenze, Italy

    Rolando Magnanini

  2. Department of Mathematical Sciences, Faculty of Science Ehime University, 2-5 Bunkyo-cho Matsuyama-shi, 790-8577, Ehime, Japan

    Shigeru Sakaguchi

Authors
  1. Rolando Magnanini
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  2. Shigeru Sakaguchi
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Correspondence to Rolando Magnanini.

Additional information

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.

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Magnanini, R., Sakaguchi, S. Stationary critical points of the heat flow in the plane. J. Anal. Math. 88, 383–396 (2002). https://doi.org/10.1007/BF02786582

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  • DOI: https://doi.org/10.1007/BF02786582

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