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, theta) near the crack tip all have the form r(k+1/2)phi(theta) with k greater than or equal to 0 integer, with the possible exception of a finite number of singularities of the form r(k) log r phi(theta). 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 1/4 + ieta + k/2 with real 77 and integer k. This is valid for general anisotropic elasticity too.