Critical points of the regular part of the harmonic Green function with Robin boundary condition

In this paper we consider the Green function for the Laplacian in a smooth bounded domain Ω ⊂ RN with Robin boundary condition ∂Gλ ∂ν +λb(x)Gλ = 0, on ∂Ω, and its regular part Sλ(x, y), whereb >0 is smooth.We show that in general, as λ→∞, the Robin function Rλ(x) = Sλ(x, x) has at least 3 critical points. Moreover, in the case b ≡ const we prove that Rλ has critical points near non-degenerate critical points of the mean curvature of the boundary, and when b ≡ const there are critical points of Rλ near non-degenerate critical points of b. © 2008 Elsevier Inc. All rights reserved