Nonlinear diffusion with a bounded stationary level surface
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 3, p. 937-952

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 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 N .

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 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 N .

DOI : https://doi.org/10.1016/j.anihpc.2009.12.001
Classification:  35K60,  35B40,  35B25
Keywords: Nonlinear diffusion equation, Overdetermined problems, Stationary level surfaces
@article{AIHPC_2010__27_3_937_0,
     author = {Magnanini, Rolando and Sakaguchi, Shigeru},
     title = {Nonlinear diffusion with a bounded stationary level surface},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {27},
     number = {3},
     year = {2010},
     pages = {937-952},
     doi = {10.1016/j.anihpc.2009.12.001},
     zbl = {1194.35209},
     mrnumber = {2629887},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2010__27_3_937_0}
}
Magnanini, Rolando; Sakaguchi, Shigeru. Nonlinear diffusion with a bounded stationary level surface. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 3, pp. 937-952. doi : 10.1016/j.anihpc.2009.12.001. http://www.numdam.org/item/AIHPC_2010__27_3_937_0/

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