Nonlinear diffusion with a bounded stationary level surface

Magnanini, Rolando; Sakaguchi, Shigeru

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 3, pp. 937-952.

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 ${ℝ}^N$ 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 ${𝕊}^N$ et l'espace hyperbolique ${ℍ}^N$.

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 ${ℝ}^N$ 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 ${𝕊}^N$ and the hyperbolic space ${ℍ}^N$.

Classification : 35K60, 35B40, 35B25
Mots clés : Nonlinear diffusion equation, Overdetermined problems, Stationary level surfaces

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