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

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.

Upload of the published version, including a minor correction of Proposition 8.1

References in corpus (4)

• Non-Autonomous Maximal $L^p$-Regularity under Fractional Sobolev Regularity in Time
• Maximal parabolic regularity for divergence operators including mixed boundary conditions
• Holomorphic families of forms, operators and $C_0$-semigroups
• Hölder estimates for parabolic operators on domains with rough boundary

Cited by in corpus (7)

• On p-elliptic divergence form operators and holomorphic semigroups
• The Kato Square Root Problem on locally uniform domains
• Euclidean structures and operator theory in Banach spaces
• $L^p$-estimates for the square root of elliptic systems with mixed boundary conditions II
• Extendability of functions with partially vanishing trace
• On the local in time well-posedness of an elliptic-parabolic ferroelectric phase-field model
• Extrapolated Elliptic Regularity and Application to the van Roosbroeck system of Semiconductors