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 ${ℍ}^N$.

[1] A.D. Aleksandrov, Uniqueness theorems for surfaces in the large V, Vestnik Leningrad Univ. 13 no. 19 (1958), 5-8, Amer. Math. Soc. Transl. Ser. 2 21 (1962), 412-415 | Zbl

[2] M.G. Crandall, H. Ishii, P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), 1-67 | Zbl

[3] L.C. Evans, H. Ishii, A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities, Ann. Inst. H. Poincaré 2 (1985), 1-20 | EuDML | Numdam | MR | Zbl

[4] L.C. Evans, P.E. Souganidis, A PDE approach to certain large deviation problems for systems of parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire S6 (1989), 229-258 | EuDML | Numdam | Zbl

[5] M.I. Freidlin, A.D. Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, New York (1984) | MR | Zbl

[6] B.H. Gilding, Hölder continuity of solutions of parabolic equations, J. London Math. Soc. 13 (1976), 103-106 | MR | Zbl

[7] S. Koike, An asymptotic formula for solutions of Hamilton–Jacobi–Bellman equations, Nonlinear Anal. 11 (1987), 429-436 | MR | Zbl

[8] S. Kumaresan, J. Prajapat, Serrin's result for hyperbolic space and sphere, Duke Math. J. 91 (1998), 17-28 | MR | Zbl

[9] P.L. Lions, P.E. Souganidis, J.L. Vázquez, The relation between the porous medium and the eikonal equations in several space dimensions, Rev. Mat. Iberoamericana 3 (1987), 275-310 | EuDML | MR | Zbl

[10] R. Magnanini, S. Sakaguchi, The spatial critical points not moving along the heat flow, J. Anal. Math. 71 (1997), 237-261 | MR | Zbl

[11] R. Magnanini, S. Sakaguchi, The spatial critical points not moving along the heat flow II: The centrosymmetric case, Math. Z. 230 (1999), 695-712, Math. Z. 232 (1999), 389 | MR | Zbl

[12] R. Magnanini, S. Sakaguchi, Matzoh ball soup: Heat conductors with a stationary isothermic surface, Ann. of Math. 156 (2002), 931-946 | MR | Zbl

[13] R. Magnanini, S. Sakaguchi, Interaction between degenerate diffusion and shape of domain, Proc. Royal Soc. Edinburgh Sect. A 137 (2007), 373-388 | MR | Zbl

[14] R. Magnanini, S. Sakaguchi, Stationary isothermic surfaces for unbounded domains, Indiana Univ. Math. J. 56 (2007), 2723-2738 | MR | Zbl

[15] J.R. Norris, Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds, Acta Math. 179 (1997), 79-103 | MR | Zbl

[16] W. Reichel, Radial symmetry for elliptic boundary-value problems on exterior domains, Arch. Rational Mech. Anal. 137 (1997), 381-394 | MR | Zbl

[17] R.C. Reilly, Mean curvature, the Laplacian, and soap bubbles, Amer. Math. Monthly 89 (1982), 180-188 | MR | Zbl

[18] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304-318 | MR | Zbl

[19] B. Sirakov, Symmetry for exterior elliptic problems and two conjectures in potential theory, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), 135-156 | EuDML | Numdam | MR | Zbl

[20] T. Strömberg, The Hopf–Lax formula gives the unique viscosity solution, Differential Integral Equations 15 (2002), 47-52 | MR | Zbl

[21] S.R.S. Varadhan, On the behavior of the fundamental solution of the heat equation with variable coefficients, Comm. Pure Appl. Math. 20 (1967), 431-455 | MR | Zbl