## Crack singularities for general elliptic systems

*
Martin Costabel,
Monique Dauge
*

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 all behave as half-integer powers of the distance *r* to the crack tip, with the
possible exception of a finite number of singularities behaving as the product of an integer power of *r* and a logarithm of *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 real part of the exponents of singularity have the form 1/4 + *k*/2 with
integer *k*. This is valid for general anisotropic elasticity too.

Dec. 1999. Preprint IRMAR 00-14.

Published in *Math. Nach.* **235**, 29-49 (2002).