We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with bounded boundary of class . 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 -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 of some substance whose density is initially a characteristic function of the complement of a domain with bounded boundary, and obtain similar results. These results are also extended to the heat flow in the sphere and the hyperbolic space .
Nous considérons la diffusion non linéaire d'une substance dans un récipient (pas nécessairement borné) avec frontière bornée de classe . 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 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 d'une substance dont la densité est initialement une fonction caractéristique du complémentaire d'un domaine ayant la frontière bornée et , et nous obtenons des résultats semblables. Ces résultats sont aussi généralisés au cas du flux de chaleur dans la sphère et l'espace hyperbolique .
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}, pages = {937--952}, publisher = {Elsevier}, volume = {27}, number = {3}, year = {2010}, doi = {10.1016/j.anihpc.2009.12.001}, mrnumber = {2629887}, zbl = {1194.35209}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2009.12.001/} }
TY - JOUR AU - Magnanini, Rolando AU - Sakaguchi, Shigeru TI - Nonlinear diffusion with a bounded stationary level surface JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 937 EP - 952 VL - 27 IS - 3 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2009.12.001/ DO - 10.1016/j.anihpc.2009.12.001 LA - en ID - AIHPC_2010__27_3_937_0 ER -
%0 Journal Article %A Magnanini, Rolando %A Sakaguchi, Shigeru %T Nonlinear diffusion with a bounded stationary level surface %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 937-952 %V 27 %N 3 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2009.12.001/ %R 10.1016/j.anihpc.2009.12.001 %G en %F 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://archive.numdam.org/articles/10.1016/j.anihpc.2009.12.001/
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