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.