Nonlinear diffusion with a bounded stationary level surface

Magnanini, Rolando ; Sakaguchi, Shigeru

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010), p. 937-952 / Harvested from Numdam

Résumé

Nous considérons la diffusion non linéaire d'une substance dans un récipient (pas nécessairement borné) avec frontière bornée de classe $ {C}^{2}$. Supposons qu'initialement, le récipient soit vide et, à sa frontière, la densité de la substance soit gardée à tout moment égale à 1. Nous montrons que, si le récipient contient un sous-domaine $ {C}^{2}$ propre à la frontière duquel la substance est gardée à tout moment à densité constante, alors la frontière du récipient doit être une sphère. Nous considérons aussi la diffusion non linéaire dans tout $ {\mathbb{R}}^{N}$ d'une substance dont la densité est initialement une fonction caractéristique du complémentaire d'un domaine ayant la frontière bornée et $ {C}^{2}$, et nous obtenons des résultats semblables. Ces résultats sont aussi généralisés au cas du flux de chaleur dans la sphère $ {\mathbb{S}}^{N}$ et l'espace hyperbolique $ {\mathbb{H}}^{N}$.

Abstract

We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with bounded boundary of class $ {C}^{2}$. Suppose that, initially, the container is empty and, at all times, the substance at its boundary is kept at density 1. We show that, if the container contains a proper $ {C}^{2}$-subdomain on whose boundary the substance has constant density at each given time, then the boundary of the container must be a sphere. We also consider nonlinear diffusion in the whole $ {\mathbb{R}}^{N}$ of some substance whose density is initially a characteristic function of the complement of a domain with bounded $ {C}^{2}$ boundary, and obtain similar results. These results are also extended to the heat flow in the sphere $ {\mathbb{S}}^{N}$ and the hyperbolic space $ {\mathbb{H}}^{N}$.

Détail

Publié le : 2010-01-01

MR 2629887 | Zbl 1194.35209

DOI : https://doi.org/10.1016/j.anihpc.2009.12.001
Classification:  35K60,  35B40,  35B25

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