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 Omega subset of R-N with Robin boundary condition partial derivative G(lambda)/partial derivative nu + lambda b(x)G(lambda) = 0, on partial derivative Omega,

and its regular part S-lambda(x,y), where b > 0 is smooth. We show that in general, as lambda -> infinity, the Robin function R-lambda(x) = S-lambda (x, x) has at least 3 critical points. Moreover, in the case b equivalent to const we prove that R-lambda has critical points near non-degenerate critical points of the mean curvature of the boundary, and when b not equivalent to const there are critical points of R-lambda near non-degenerate critical points of b.