arXiv subjects

A. Driouich

Publications and source records attributed to A. Driouich.

5 recordsLinked to original sources

arXiv2020

Staffans-Weiss perturbations for Maximal $L^p$-regularity in Banach spaces

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.

math.FA↗

arXiv2020

On the maximal regularity for a class of Volterra integro-differential equations

We propose an approach based on perturbation theory to establish maximal $L^p$-regularity for a class of integro-differential equations. As the left shift semigroup is involved for such equations, we study maximal regularity on Bergman spaces for autonomous and non-autonomous integro-differential equations. Our method is based on the formulation of the integro-differential equations to a Cauchy problems, infinite dimensional systems theory and some recent results on the perturbation of maximal regularity (see \cite{AmBoDrHa}). Applications to heat equations driven by the Dirichlet (or Neumann)-Laplacian are considered.

math.FA↗

arXiv2018

On norm continuity, differentiability and compactness of perturbed semigroups

The main purpose of this paper is to treat semigroups properties, like norm continuity, compactness and differentiability for perturbed semigroups in Banach spaces. In particular, we investigate three large classes of perturbations, Miyadera-Voigt, Desch-Schappacher and Staffans-Weiss perturbations. Our approach is mainly based on feedback theory of Salamon-Weiss systems. Our results are applied to abstract boundary integro-differential equations in Banach spaces.

math.FA↗

arXiv2018

Maximal $L^p$-regularity for perturbed evolution equations in Banach spaces

The main purpose of this paper is to investigate the concept of maximal $L^p$-regularity for perturbed evolution equations in Banach spaces. We mainly consider three classes of perturbations: Miyadera-Voigt perturbations, Desch-Schappacher perturbations, and more general Staffans-Weiss perturbations. We introduce conditions for which the maximal $L^p$-regularity can be preserved under these kind of perturbations. We give examples for a boundary perturbed heat equation in $L^r$-spaces and a perturbed boundary integro-differential equation. We mention that our results mainly extend those in the works: [P. C. Kunstmann and L. Weis, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), 415-435] and [B.H. Haak, M. Haase, P.C. Kunstmann, Adv. Differential Equations 11 (2006), no. 2, 201-240].

math.FA↗

arXiv2007

Spectral decomposition and Gelfand's theorem

In this paper we are interested in spectral decomposition of an unbounded operator with discrete spectrum. We show that if $A$ generates a polynomially bounded $n$-times integrated group whose spectrum set $σ(A)=\{iλ_k; k\in\mathbb{Z}^* \}$ is discrete and satisfies $\sum \frac{1}{|λ_k|^\ellδ_k^n}<\infty$ ($n$ and $\ell$ nonnegative integers), then there exists projectors $(P_k)_{k\in\mathbb{Z}^*}$ such that $\sum P_kx=x$ ($ x\in D(A^{n+\ell})$), where $δ_k=\min(\frac{| λ_{k+1}-λ_k|}2, \frac{|λ_{k-1}-λ_k|}2)$.

math.SP↗