2011/01/20 by Felix Schlenk, Pieralberto Sicbaldi, Schlenk, Felix +1 · 5 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1101.3988
openalex publication_date 2011/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove the existence of a smooth family of non-compact domains Omegas ⊂ Rn+1 bifurcating from the straight cylinder Bn × R for which the first eigenfunction of the Laplacian with 0 Dirichlet boundary condition also has constant Neumann data at the boundary. The domains Omegas are rotationally symmetric and periodic with respect to the R-axis of the cylinder; they are of the form Omegas = (x,t) ∈ Rn × R | |x| < 1+s cos((2π)/Ts t) + O(s2) where Ts = T0 + O(s) and T0 is a positive real number depending on n. For n ≥ 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 T0. This work improves a recent result of the second author.