Maximal $L^p$-regularity and $H^{\infty}$-calculus for block operator matrices and applications

TL;DR

This work develops a comprehensive operator-theoretic framework for 2×2 block operator matrices with diagonal dominance, establishing sectoriality, $\mathcal{R}$-sectoriality, and a bounded $H^{\infty}$-calculus for the full matrix from the diagonal part. Central to the approach is a resolvent-factorization $\lambda-\mathcal{A}=\mathcal{M}(\lambda)(\lambda-\mathcal{D})$ and the analysis of Schur complements $M_1(\lambda),M_2(\lambda)$, which allows perturbation results for large but structured couplings without classical smallness assumptions. The authors also develop extrapolation results across consistent scales and provide a range of applications to parabolic and damped-wave–type systems arising in liquid crystals, Keller–Segel chemotaxis, and Stokes-type models, yielding maximal $L^p_t$-regularity and well-posedness for linearized problems and their nonlinear counterparts. Overall, the paper offers a robust framework for coupled PDEs with mixed orders, enabling precise regularity and calculus results that extend to various complex physical models and extrapolation regimes.

Abstract

Many coupled evolution equations can be described via $2\times2$-block operator matrices of the form $\mathcal{A}=\begin{bmatrix} A & B \\ C & D \end{bmatrix}$ in a product space $X=X_1\times X_2$ with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator $\mathcal{A}$ can be seen as a relatively bounded perturbation of its diagonal part with $\mathsf{D}(\mathcal{A})=\mathsf{D}(A)\times \mathsf{D}(D)$ though with possibly large relative bound. For such operators the properties of sectoriality, $\mathcal{R}$-sectoriality and the boundedness of the $H^\infty$-calculus are studied, and for these properties perturbation results for possibly large but structured perturbations are derived. Thereby, the time dependent parabolic problem associated with $\mathcal{A}$ can be analyzed in maximal $L^p_t$-regularity spaces, and this is applied to a wide range of problems such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model.

Theorems & Definitions (87)

• Proposition 2.1: $H^{\infty}$-calculus for shifted operators
• proof
• Remark 3.2: Boundedness of $B(\lambda-D)^{-1}$ and $C(\lambda-A)^{-1}$
• Proposition 3.3: Factorization of diagonally dominant $\mathcal{A}$
• proof : Proof of Proposition \ref{['l:relation_resolvent_AD_calA']}
• Example 3.4: The condition on $(\lambda-A)^{-1}B$
• Theorem 4.1: Characterization of sectoriality and $\mathcal{R}$-sectoriality
• Remark 4.2: Optimality of the angle
• Remark 4.3: Closedness of $\mathcal{A}$
• Corollary 4.4: Characterization of sectoriality and $\mathcal{R}$-sectoriality for invertible $\mathcal{D}$