  We show in this paper a Lipschitz stability result for a crack inverse
problem in half space. The direct problem is a Laplace equation with zero
Neumann condition on the top boundary. The forcing term is a discontinuity
across the crack. This formulation can be related to geological faults in
elastic media or to irrotational incompressible flows in a half space minus an
inner wall. The direct problem is well posed in an appropriate functional
space. We study the related inverse problem where the jump across the crack is
unknown, and more importantly, the geometry and the location of the crack are
unknown. The data for the inverse problem is of Dirichlet type over a portion
of the top boundary. We prove that this inverse problem is uniquely solvable
under some assumptions on the geometry of the crack. The highlight of this
paper is showing a stability result for this inverse problem. Assuming that the
crack is planar, we show that reconstructing the plane containing the crack is
Lipschitz stable despite the fact that the forcing term for the underlying PDE
is unknown. This uniform stability result holds under the assumption that the
forcing term is bounded above and the Dirichlet data is bounded below away from
zero in appropriate norms.
