Bifurcating extremal domains for the first eigenvalue of the Laplacian

Abstract

We prove the existence of a smooth family of non-compact domains $Omeg{a}_s\subset R^{n+1}$ 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 $Omeg{a}_s$ are rotationally symmetric and periodic with respect to the R-axis of the cylinder; they are of the form $Omeg{a}_s=(x,t)\in R^n\times R\mid |x|<1+s\cos ((2\pi )/T_{s}t)+O(s^2)$ 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.

28 pages, 3 figures