  We extend in two directions the notion of perturbations of Carleson type for
the Dirichlet problem associated to an elliptic real second-order
divergence-form (possibly degenerate, not necessarily symmetric) elliptic
operator. First, in addition to the classical perturbations of Carleson type,
that we call additive Carleson perturbations, we introduce
scalar-multiplicative and antisymmetric Carleson perturbations, which both
allow non-trivial differences at the boundary. Second, we consider domains
which admit an elliptic PDE in a broad sense: we count as examples the 1-sided
NTA (a.k.a. uniform) domains satisfying the capacity density condition, the
1-sided chord-arc domains, the domains with low-dimensional Ahlfors-David
regular boundaries, and certain domains with mixed-dimensional boundaries; thus
our methods provide a unified perspective on the Carleson perturbation theory
of elliptic operators.
  Our proofs do not introduce sawtooth domains or the extrapolation method. We
also present several applications to some Dahlberg-Kenig-Pipher operators,
free-boundary problems, and we provide a new characterization of $A_{\infty}$
among elliptic measures.
