On the Persistence of the Multiplicity of Eigenvalues for Some Variational Elliptic Operator Depending on the Domain

Given a bounded domain Ω ⊂ of Rm and an eigenvalue λ* of multiplicity 2 for a variational elliptic operator L, with Dirichtet or Neumann boundary conditions, we show that the set of C2k+1(Ω) perturbations of Ω, which preserves the multiplicity of the eigenvalue on the perturbed domain, is a manifold of codimension 2 in C2k+1(Ω).