Maximal Lp-regularity and H∞-calculus for block operator matrices and applications

Many coupled evolution equations can be described via 2 x 2-block operator matrices of the form *A=**A**B**C**D** * in a product space X = X1 x X2 with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator *A* can be seen as a relatively bounded perturbation of its diagonal part though with possibly large relative bound. For such operators the properties of sectoriality, R-sectoriality and the boundedness of the *H**∞* -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 *A* can be analyzed in maximal *L**t**p* -regularity spaces, and this is illustrated by a number of applications such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model. The approach developed here is based in spirit on a combination of the theory by Kalton, Kunstmann and Weis (Perturbation and interpolation theorems for the *H**∞* -calculus with applications to differential operators. Math. Ann., 336(4):74-801, 2006) relating R-sectoriality and the boundedness of the *H**∞* -calculus with concepts for diagonally dominant block operator matrices pioneered by Nagel (Towards a "matrix theory" for unbounded operator matrices. Math. Z., 201(1):57-68, 1989) for C0-semigroups.

The presentation is based on a joint work with Antonio Agresti, see doi.org/10.1016/j.jfa.2023.110146.