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Matzoh ball soup revisited: the boundary regularity issue
Author(s) -
Magnanini Rolando,
Sakaguchi Shigeru
Publication year - 2013
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1551
Subject(s) - mathematics , ball (mathematics) , bounded function , mathematical analysis , boundary value problem , initial value problem , cauchy distribution , domain (mathematical analysis) , heat equation , boundary (topology) , nonlinear system , cauchy boundary condition , cauchy problem , mixed boundary condition , physics , quantum mechanics
We consider nonlinear diffusion equations of the form ∂ t u = Δ ϕ ( u ) inR Nwith N ≥ 2. When ϕ ( s ) ≡ s , this is just the heat equation. Let Ω be a domain inR N , where ∂ Ω is bounded and ∂Ω = ∂R N ∖ Ω ¯. We consider the initial‐boundary value problem, where the initial value equals zero and the boundary value equals 1, and the Cauchy problem where the initial data is the characteristic function of the setΩ c = R N ∖ Ω . We settle the boundary regularity issue for the characterization of the sphere as a stationary level surface of the solution u :, no regularity assumption is needed for ∂ Ω. Copyright © 2012 John Wiley & Sons, Ltd.