The Kato square root problem for mixed boundary value problems
  We solve the Kato square root problem for second order elliptic systems in divergence form under mixed boundary conditions on Lipschitz domains. This answers a question posed by J.-L. Lions in 1962. To do this we develop a general theory of quadratic estimates and functional calculi for complex perturbations of Dirac-type operators on Lipschitz domains. 
Andreas Axelsson, Stephen Keith and Alan McIntosh