Singularities of non-rotationally symmetric solutions of boundary value problems for the Lamé equations in a 3 dimensional domain with conical points

Abstract

It is well known that singularities are present in solutions of boundary value problems for the Lamé equations in conical domains. It follows from the general theory that the solutions consist of singular terms of the form r α (ln r) q F(α, φ, θ) (r is the distance to the vertex of the cone φ, and θ are the spherical angles) and α more regular term. Rotationally symmetric solutions of the Lamé equations under zero boundary displacements or stress free boundary conditions are investigated in (1, 2), where the values of α and q have been computed. Here we are concerned with the more general case, namely that the volume and surface forces of our problems are non rotationally symmetric. That means that the solutions depend not only on r and θ, but on the polar angle φ too. Using a monotonicity principle of Kozlov, Maz'ja und Schwab one can get regularity results for polyhedral domains too.