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Interaction between degenerate diffusion and shape of domain

Published online by Cambridge University Press:  26 March 2007

R. Magnanini  and
S. Sakaguchi
R. Magnanini
Affiliation:
Dipartimento di Matematica U. Dini, Università di Firenze, viale Morgagni 67/3A, 50134 Firenze, Italy (magnanin@math.unifi.it)
S. Sakaguchi
Affiliation:
Department of Mathematics, Faculty of Science, Ehime University, 2-5 Bunkyo-cho, Matsuyama-shi, Ehime 790-8577, Japan (sakaguch@dpc.ehime-u.ac.jp)
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Abstract

We consider the flow of a gas into a bounded tank Ω with smooth boundary ∂Q. Initially Ω is empty, and at all times the density of the gas is kept constant on ∂Ω. Choose a number R > 0 sufficiently small that, for any point x in Q having distance R from ∂Ω, the closed ball B with radius R centred at x intersects ∂Ω at only one point. We show that if the gas content of such balls B is constant at each given time, then the tank Ω must be a ball. In order to prove this, we derive an asymptotic estimate for gas content for short times. Similar estimates are also derived in the case of the evolution p-Laplace equation for p ⩾ 2.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 137 , Issue 2 , 2007 , pp. 373 - 388
Copyright
Copyright © Royal Society of Edinburgh 2007

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