  We study the minimization of anisotropic total variation functionals with
additional measure terms among functions of bounded variation subject to a
Dirichlet boundary condition. More specifically, we identify and characterize
certain isoperimetric conditions, which prove to be sharp assumptions on the
signed measure data in connection with semicontinuity, existence, and
relaxation results. Furthermore, we present a variety of examples which
elucidate our assumptions and results.
