  We study the factorization and monotonicity method for inverse acoustic
scattering problems. Firstly, we newly give the general functional analysis
theorem for monotonicity method. Comparing with the factorization method, the
general theorem of the monotonicity generates reconstruction schemes under
weaker a priori assumptions for unknown targets, and directly can deal with
mixed problems that unknown targets have several different boundary conditions.
Using the general theorem, we give the reconstruction scheme for the mixed
crack that the Dirichlet boundary condition is imposed on one side of the crack
and the Neumann boundary condition on the other side, which would be a new
extension of monotonicity method.
