Crack singularities for general elliptic systems

We consider general homogeneous Agmon-Douglis-Nirenberg elliptic systems with constant coefficients complemented by the same set of boundary conditions on both sides of a crack in a two-dimensional domain. We prove that the singular functions expressed in polar coordinates (r; ) near the crack tip all have the form r 12+k'( ) with k 0 integer, with the possible exception of a finite number of singularities of the form rk log r '( ) . We also prove results about singularities in the case when the boundary conditions on the two sides of the crack are not the same, and in particular in mixed Dirichlet-Neumann boundary value problems for strongly coercive systems: in the latter case, we prove that the exponents of singularity have the form 14 + i + k2 with real and integer k . This is valid for general anisotropic elasticity too.