  In this paper, we consider the direct and inverse problem for isotropic
scatterers with two conductive boundary conditions. First, we show the
uniqueness for recovering the coefficients from the known far-field data at a
fixed incident direction for multiple frequencies. Then, we address the inverse
shape problem for recovering the scatterer for the measured far-field data at a
fixed frequency. Furthermore, we examine the direct sampling method for
recovering the scatterer by studying the factorization for the far-field
operator. The direct sampling method is stable with respect to noisy data and
valid in two dimensions for partial aperture data. The theoretical results are
verified with numerical examples to analyze the performance by the direct
sampling method.
