Lp-estimates for the square root of elliptic systems with mixed boundary conditions

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∈(p0,2) to an isomorphism between a closed subspace of W1,p 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∞-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.