The Laplace and the linear elasticity problems near polyhedral corners and associated eigenvalue problems

For domains with concave corners, the solutions to elliptic boundary values have the typical r(alpha)-singularity. The so-called singularity exponents (alpha)are the eigenvalues of an eigenvalue problem which is associated with the given boundary value problem. This paper is aimed at deriving the mentioned eigenvalue problems for two examples, the Laplace equation and the linear elasticity problem. We will show interesting properties of these eigenvalue problems. For the linear elasticity problem, we explain in addition why the classical symmetry and positivity assumptions of the material tensor have to be used with care.