## Asymptotics without logarithmic terms for crack problems

*
Martin Costabel,
Monique Dauge,
Roland Duduchava
*

We consider boundary value problems for elliptic systems in a domain complementary to a
smooth surface *M* with boundary *E*. The same boundary conditions are prescribed on both
sides of the surface *M*. The most important model behind this
investigation is the crack problem in three-dimensional linear elasticity
(either isotropic or anisotropic): there the boundary conditions are Neumann, ie tractions are prescribed on both faces of the
crack surface *M*. We prove that the singular functions appearing in the
expansion of the solution along the crack edge *E* all have the form
*r*^(k+1/2) ψ(θ) in local polar
coordinates (*r*,θ) : The logarithmic shadow terms predicted by the general theory do
not appear. Moreover, we obtain that, for a smooth right
hand side, the jump of the displacement across the crack surface
is the product of *r*^(1/2) with a smooth vector function on *M*.

We present two different, but complementing, approaches leading to these
results, and providing distinct generalizations. The first one is based on a
Wiener-Hopf factorization of the pseudodifferential symbol on the
surface *M* obtained after reduction of the boundary value problem.
The condition on the symbol which yields the absence of logarithmic terms into
the solution of the boundary pseudodifferential equation
is a variant of the transmission condition.
The asymptotics of the solution in the full space is then deduced by a representation formula from the asymptotics of the solution on *M*.
The second approach concerns directly the boundary value problem and
is based on a closer look at the Mellin symbol at each point of the crack edge *E*. The Mellin symbol is proved to act between special subspaces of angular functions and the absence of logarithmic terms is the consequence of a series of compatibility conditions, valid for any Agmon--Douglis--Nirenberg system.

October. 2001. Preprint IRMAR 01-50.

*Comm. P.D.E.* **28**, no 5 & 6, 2003, 869-926.