  We analyze a mathematical model of elastic dislocations with applications to
geophysics, where by an elastic dislocation we mean an open, oriented Lipschitz
surface in the interior of an elastic solid, across which there is a
discontinuity of the displacement. We model the Earth as an infinite,
isotropic, inhomogeneous, elastic medium occupying a half space, and assume
only Lipschitz continuity of the Lam\'e parameters. We study the well posedness
of very weak solutions to the forward problem of determining the displacement
by imposing traction-free boundary conditions at the surface, continuity of the
traction and a given jump on the displacement across the fault. We employ
suitable weighted Sobolev spaces for the analysis. We utilize the well
posedness of the forward problem and unique-continuation arguments to establish
uniqueness in the inverse problem of determining the dislocation surface and
the displacement jump from measuring the displacement at the surface of the
Earth. Uniqueness holds for tangential or normal jumps and under some geometric
conditions on the surface.
