A generic property for the eigenfunctions of the Laplacian (Q1411248)

The authors show that generically with respect to \(C^2\)-bounded regions \(\Omega\) in \(\mathbb R^n\), \(n\geq 2\), the inequality \(\int_\Omega \phi^3\neq 0\) holds for any eigenfunction of the Laplacian under Dirichlet or Neumann boundary condition. The motivation for the result comes from an (unspecified) stability problem in reaction-diffusion systems. A transversality theorem and boundary perturbation calculus developed by D. Henry are used as the main tools in the proofs.