Killed distribution dependent SDE for nonlinear Dirichlet problem

To characterize nonlinear Dirichlet problems in an open domain, we investigate killed distribution dependent SDEs. By constructing the coupling by projection and using Zvonkin/Girsanov transforms, well-posedness is proved for three different situations:1) monotone case with noise (possibly degenerate);2) singular non-degenerate noise;3) independent noise. In first two cases domain $ C^2 smooth such that Lipschitz continuity initial distributions also derived, last arbitrary.