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

Abstract

This article focuses on L p -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 L p 0 , then we prove that the square root extends for all p ∈ (p0, 2) to an isomorphism between a closed subspace of W 1,p carrying the boundary conditions and L p . 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 L p . 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 L 2 for systems with lower order terms in our setting.

Stochastic maximal regularity for rough time-dependent problems

“…• [38]: second order elliptic systems in divergence form on bounded Lipschitz domains, with L ∞ coefficients and mixed boundary conditions. • [76]: Stokes operator on a Lipschitz domain • [79]: Dirichlet Laplace operator on C 2 -domains with weights • [86]: Hodge Laplacian and Stokes operator with Hodge boundary conditions on very weakly Lipschitz domains One advantage of the above approach is that it leads to an L p ( × (0, T ); L q )-theory for all p ∈ (2, ∞) and q ∈ [2, ∞) (where in case q = 2, the case p = 2 is included), and gives optimal space-time regularity results such as U ∈ L p ( ; H θ, p (0, T ; X 1−θ )) or even U ∈ L p ( ; C([0, T ]; X 1− 1 p , p )), where we used complex and real interpolation space respectively.…”

On p-elliptic divergence form operators and holomorphic semigroups

“…An independent interest in Theorem 1 stems from its immediate consequences for the harmonic analysis of L on L q (O), such as H ∞ -calculus, Riesz transforms and Kato square root estimates, at least when O is either the whole space or a bounded connected set that is Lipschitz regular around the Neumann boundary part. Indeed, such results come for free once the extrapolation of the semigroup has been settled and we refer the reader to [3,9] for a precise account.…”

On the Lp-theory of C0-semigroups associated with second-order elliptic operators with complex singular coefficients

“…Assumptions Ω c and D are satisfied, the same estimate holds for all p ∈ [2, 2 + ε), where ε > 0 is as in Theorem 3.4(b). Moreover, there exists a C > 0 such that In [25], Egert proved under the assumptions that Ω is a d-set and D a (d − 1)-set that there exists an ε > 0 such that for all numbers p that satisfy (15) or (13) the operator A…”

“…The isomorphism property of (25) is deduced in [4] for real coefficient functions µ, while in his pioneering paper [25] Egert succeeded to prove the isomorphy (25) for complex coefficient functions (and even for systems) as long as the corresponding semigroup is well-behaved on L p (Ω). In order to go into details, we quote and apply the results from [25] that are relevant for our purposes. In all what follows, for a given coefficient function µ and p ∈ ( p 0 (µ)d d(p 0 (µ)−1)+2 , p 0 (µ)d d−2 ) we denote by A µ p the operator corresponding to µ.…”

“…This considerably generalizes the results in [35,Thm. 5.16 and Section 6] in view of complex coefficients and much more general admissible geometries for Ω and D. • If one is only interested in maximal parabolic regularity, it is also possible to transport this property directly from L q (Ω) to W −1,q D (Ω) by the square root isomorphism (25), see [36,Lem. 5.12].…”