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