Polygonal heat conductors with a stationary hot spot |
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Authors: | Rolando Magnanini and Shigeru Sakaguchi |
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Affiliation: | (1) Dipartimento di Matematica U. Dini, Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy;(2) Department of Mathematics, Graduate School of Science and Engineering, Ehime University, 2-5 Bunkyo-cho, Matsuyama-shi, Ehime 790-8577, Japan;(3) Present address: Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan |
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Abstract: | We consider a convex polygonal heat conductor whose inscribed circle touches every side of the conductor. Initially, the conductor has constant temperature and, at every time, the temperature of its boundary is kept at zero. The hot spot is the point at which temperature attains its maximum at each given time. It is proved that, if the hot spot is stationary, then the conductor must satisfy two geometric conditions. In particular, we prove that these geometric conditions yield some symmetries provided the conductor is either pentagonal or hexagonal. This research was partially supported by Grants-in-Aid for Scientific Research (B) (# 12440042) and (B) (# 15340047) of Japan Society for the Promotion of Science, and by a Grant of the Italian MURST. |
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