  This paper focuses on an elastic dislocation problem that is motivated by
applications in the geophysical and seismological communities. In our model,
the displacement satisfies the Lam\'e system in a bounded domain with a mixed
homogeneous boundary condition. We also allow the occurrence of discontinuities
in both the displacement and traction fields on the fault curve/surface. By the
variational approach, we first prove the well-posedness of the direct
dislocation problem in a rather general setting with the Lam\'e parameters
being real-valued $L^\infty$ functions and satisfy the strong convexity
condition. Next, by considering the scenario that the Lam\'e parameters are
constant and the fault curve/surface possesses certain corner singularities, we
establish a local characterisation of the slip vectors at the corner points
over the dislocation curve/surface. In our study the dislocation is
geometrically rather general and may be open or closed. For both cases, we
establish the uniqueness results for the inverse problem of determining the
dislocation curve/surface and the slips.
