Bifurcating extremal domains for the first eigenvalue of the Laplacian

We prove the existence of a smooth family of non-compact domains Ωs⊂Rn+1Ωs⊂Rn+1, n⩾1n⩾1, bifurcating from the straight cylinder Bn×RBn×R for which the first eigenfunction of the Laplacian with 0 Dirichlet boundary condition also has constant Neumann data at the boundary: For each s∈(−ε,ε)s∈(−ε,ε), the overdetermined system{Δu+λu=0in Ωs,u=0on ∂Ωs,〈∇u,ν〉=conston ∂Ωs has a bounded positive solution. The domains ΩsΩs are rotationally symmetric and periodic with respect to the RR-axis of the cylinder; they are of the formΩs={(x,t)∈Rn×R|‖x‖<1+scos(2πTst)+O(s2)} where Ts=T0+O(s)Ts=T0+O(s) and T0T0 is a positive real number depending on n  . For n⩾2n⩾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 T0T0. This work improves a recent result of the second author.