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

Abstract

Many coupled evolution equations can be described via $2\times 2$-block operator matrices of the form $\mathcal{A}=\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]$ 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 $L_t^p$-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.

arXiv comment55 pages. Accepted for publication in JFA

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