Fourth-Order Operators on Manifolds with Boundary
  Recent work in the literature has studied fourth-order elliptic operators on manifolds with boundary. This paper proves that, in the case of the squared Laplace operator, the boundary conditions which require that the eigenfunctions and their normal derivative should vanish at the boundary lead to self-adjointness of the boundary-value problem. On studying, for simplicity, the squared Laplace operator in one dimension, on a closed interval of the real line, alternative conditions which also ensure self-adjointness set to zero at the boundary the eigenfunctions and their second derivatives, or their first and third derivatives, or their second and third derivatives, or require periodicity, i.e. a linear relation among the values of the eigenfunctions at the ends of the interval. For the first four choices of boundary conditions, the resulting one-loop divergence is evaluated for a real scalar field on the portion of flat Euclidean four-space bounded by a three-sphere, or by two concentric three-spheres. 
Giampiero Esposito and Alexander Yu. Kamenshchik