2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/100984We 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.To appear in the Journal of the London Mathematical Society. This arxiv version is distinct from the final versionAnalysis of PDEsClassical Analysis and ODEs35J25; 47B44; 47F05text