Bifurcating extremal domains for the first eigenvalue of the Laplacian

We prove the existence of a smooth family of non-compact domains $Omega_s \subset R^{n+1}$ bifurcating from the straight cylinder $B^n \times R$ for which the first eigenfunction of the Laplacian with 0 Dirichlet boundary condition also has constant Neumann data at the boundary. The domains $Omega_s$ are rotationally symmetric and periodic with respect to the R-axis of the cylinder; they are of the form $Omega_s = {(x,t) \in R^n \times R \mid |x| < 1+s \cos((2\pi)/T_s t) + O(s^2)}$ where $T_s = T_0 + O(s)$ and T_0 is a positive real number depending on n. For $n \ge 2$ these domains provide a smooth family of counter-examples to a conjecture of Berestycki, Caffarelli and Nirenberg. We also give rather precise upper and lower bounds for the bifurcation period T_0. This work improves a recent result of the second author.