We study the spectral behavior of higher order elliptic operators upon domain perturbation. We prove general spectral stability results for Dirichlet, Neumann and intermediate boundary conditions. Moreover, we consider the case of the bi-harmonic operator with those intermediate boundary conditions which appears in study of hinged plates. In this case, we analyze the spectral behavior when the boundary of the domain is subject to a periodic oscillatory perturbation. We will show that there is a critical oscillatory behavior and the limit problem depends on whether we are above, below or just sitting on this critical value. In particular, in the critical case we identify the strange term appearing in the limiting boundary conditions by using the unfolding method from homogenization theory.

Let Ω be an open connected subset of R n for which the imbedding of the Sobolev space W 1,2 (Ω) into the space L 2 (Ω) is compact. We consider the Neumann eigenvalue problem for the Laplace operator in the open subset φ(Ω) of R n , where φ is a Lipschitz continuous homeomorphism of Ω onto φ(Ω). Then we prove a result of real analytic dependence for symmetric functions of the eigenvalues upon variation of φ. Mathematics Subject Classification (2000). Primary 35P15; Secondary 47H30.

We prove sharp stability results for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Dirichlet boundary conditions upon domain perturbation. The main results concern estimates for the variation of the eigenvalues via the Hausdorff distance between the domains or the Lebesgue measure of their symmetric difference. Our analysis includes domains with Lipschitz boundaries as well as domains with boundary degenerations of power type

We consider a class of eigenvalue problems for poly-harmonic operators, including Dirichlet and buckling-type eigenvalue problems. We prove an analyticity result for the dependence of the symmetric functions of the eigenvalues upon domain perturbations and compute Hadamard-type formulas for the Frechét differentials. We also consider isovolumetric domain perturbations and characterize the corresponding critical domains for the symmetric functions of the eigenvalues. Finally, we prove that balls are critical domains.

We prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation