  This article focuses on Lp-estimates for the square root of elliptic systems
of second order in divergence form on a bounded domain. We treat complex
bounded measurable coefficients and allow for mixed Dirichlet/Neumann boundary
conditions on domains beyond the Lipschitz class. If there is an associated
bounded semigroup on Lp0 , then we prove that the square root extends for all p
$\in$ (p0, 2) to an isomorphism between a closed subspace of W1p carrying the
boundary conditions and Lp. This result is sharp and extrapolates to exponents
slightly above 2. As a byproduct, we obtain an optimal p-interval for the
bounded H$\infty$-calculus on Lp. Estimates depend holomorphically on the
coefficients, thereby making them applicable to questions of non-autonomous
maximal regularity and optimal control. For completeness we also provide a
short summary on the Kato square root problem in L2 for systems with lower
order terms in our setting.
