Abstract
In this note we consider problems of the following type: Let D be a bounded domain in Euclidean n space (R n) and suppose u is a positive solution to Lu = 0 in D with u(x) → 0, |Δu(x)| → constant, as x → ∂D, in an appropriate sense. Show that D is a ball and u is radially symmetric about the center of D. This problem was solved very elegantly by Serrin [25] under the assumption that ∂D is of class C 2, and where a, b, c, are continuously differentiable in each variable. Also L is elliptic and repeated indices denote summation from 1 to n. Immediately following Serrin’s article, Weinberger [30] gave another proof of Serrin’s theorem when Lu = Δu +1 = 0. From Weinberger’s proof it is clear that the assumption, ∂D ∈ C 2 is unnecessary in this special case, provided the boundary conditions are interpreted as, (A) Given ε > 0 there exists a neighborhood N of ∂D such that u(x) < ε, |∇u(x)|—a| < ε, when x ∈ N. In (A), a denotes a positive constant.
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Lewis, J.L., Vogel, A. (1992). On Some Almost Everywhere Symmetry Theorems. In: Lloyd, N.G., Ni, W.M., Peletier, L.A., Serrin, J. (eds) Nonlinear Diffusion Equations and Their Equilibrium States, 3. Progress in Nonlinear Differential Equations and Their Applications, vol 7. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0393-3_24
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DOI: https://doi.org/10.1007/978-1-4612-0393-3_24
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