Martin-Luther-Universität Halle-Wittenberg


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16. Januar 2024

Amru Hussein (RPTU Kaiserslautern-Landau)

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

Many coupled evolution equations can be described via 2 × 2-block operator matrices of the form \mathcal{A}=\begin{bmatrix} A & B \\ C & D \end{bmatrix} 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 \mathcal{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  \mathcal{A} can be analyzed in maximal Lpt -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   

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