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

Abstract

In this paper we consider the Green function for the Laplacian in a smooth bounded domain Ω ⊂ R N with Robin boundary condition ∂ G λ ∂ ν + λ b ( x ) G λ = 0 , on ∂ Ω , and its regular part S λ ( x , y ) , where b > 0 is smooth. We show that in general, as λ → ∞ , the Robin function R λ ( x ) = S λ ( x , x ) has at least 3 critical points. Moreover, in the case b ≡ const we prove that R λ has critical points near non-degenerate critical points of the mean curvature of the boundary, and when b ≢ const there are critical points of R λ near non-degenerate critical points of b.