On the nondegeneracy of the critical points of the Robin function in symmetric domains

Let Omega be a smooth bounded domain of R-N, N greater than or equal to 2, which is symmetric with respect to the origin. In this Note we prove that, under some geometrical condition on Omega (for example convexity in the directions x(1),...,x(N)), the Hessian matrix of the Robin function computed at zero is diagonal and strictly negative definite. (C) 2002 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.