Advances in Differential Equations

Perturbation, interpolation, and maximal regularity

Bernhard H. Haak, Markus Haase, and Peer C. Kunstmann

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We prove perturbation theorems for sectoriality and $R$--sectoriality in Banach spaces, which yield results on perturbation of generators of analytic semigroups and on perturbation of maximal $L^p$--regularity. For a given sectorial or $R$--sectorial operator $A$ in a Banach space $X$ we give conditions on intermediate spaces $Z$ and $W$ such that, for an operator $S: Z\to W$ of small norm, the perturbed operator $A+S$ is again sectorial or $R$--sectorial, respectively. These conditions are obtained by factorising the perturbation as $S= -BC$, where $B$ acts on an auxiliary Banach space $Y$ and $C$ maps into $Y$. Our results extend previous work on perturbations in the scale of fractional domain spaces associated with $A$ and allow for a greater flexibility in choosing intermediate spaces for the action of perturbation operators. At the end we illustrate our results with several examples, in particular with an application to a ``rough'' boundary-value problem.

Article information

Adv. Differential Equations, Volume 11, Number 2 (2006), 201-240.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A55: Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]
Secondary: 34G10: Linear equations [See also 47D06, 47D09] 35K90: Abstract parabolic equations 46B70: Interpolation between normed linear spaces [See also 46M35] 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]


Haak, Bernhard H.; Haase, Markus; Kunstmann, Peer C. Perturbation, interpolation, and maximal regularity. Adv. Differential Equations 11 (2006), no. 2, 201--240.

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