Remarks on the factorization and monotonicity method for inverse acoustic scatterings

We study the factorization and monotonicity method for inverse acoustic scattering problems. Firstly, we give a new general functional analysis theorem for the monotonicity method. Comparing with the factorization method, the general theorem of the monotonicity generates reconstruction schemes under weaker {\it a priori} assumptions for unknown targets, and can directly deal with mixed problems so that the 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 is a new extension of monotonicity method.