Edge asymptotics on a skew cylinder: complex variable form

Abstract

1.a. Introduction. It is well known that singularities of the domain give rise to a loss of regularity for the solutions of any elliptic boundary value problem. The situation is rather well understood when the singularities are isolated points of the boundary and are of conical type (see [5], [9]). When a conical singularity is tensorized with an affine space, one gets an edge. Regularity results are rather complete in that case [8], [13]. If the operator is translation invariant along the edge, the asymptotics can be derived in a direct way from the asymptotics on the corresponding conical domain [3]. But for physical examples in the ordinary three-dimensional space, when a bounded domain has edges and no corners, then the edges are necessarily curved. The simplest example is a cylinder with circular basis, cut orthogonally to its generating lines. But this example is very particular: The opening of the edge is everywhere π/2 and the curvature of the edge is constant. If one cuts the cylinder by a plane which is skew with respect to the generating lines, then the edge is elliptic and the opening angle is varying. This gives rise to difficulties for the precise analysis of the structure of the solution, due to the fact that the asymptotics in the corresponding two-dimensional domains depend in a discontinuous way on the opening parameter. In particular, the coefficients of the singular functions along the edge (stress intensity factors etc.) will blow up at certain points. Such a behavior causes difficulties also for numerical approximations.