  In this paper we show that the concept of maximal $L^p$-regularity is stable
under a large class of unbounded perturbations, namely Staffans-Weiss
perturbations. To that purpose, we first prove that the analyticity of
semigroups is preserved under this class of perturbations, which is a necessary
condition for the maximal regularity. In UMD spaces, $\mathcal{R}$-boundedness
conditions are exploited to give conditions guaranteing the maximal regularity.
For non-reflexive Banach space, a condition is imposed to the Dirichlet
operator associated to the boundary value problem to prove the maximal
regularity. A Pde example illustrating the theory and an application to a class
of non-autonomous perturbed boundary value problems are presented.
