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

TL;DR

This paper establishes sharp Lp-estimates for the square root of elliptic systems with mixed boundary conditions, extending known results to complex coefficients and non-Lipschitz domains, with implications for maximal regularity and control.

Contribution

It proves the boundedness and isomorphism of the square root operator on Lp spaces for a broad class of elliptic systems with mixed boundary conditions, extending previous results beyond Lipschitz domains.

Findings

Sharp Lp-estimates for elliptic systems with mixed boundary conditions.

Extension of the square root operator to a range of p values beyond 2.

Optimal p-interval for bounded H∞-calculus on Lp.

Abstract

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.