Ahmed Medeghri

On a parabolic equation in a triangular domain

We prove the optimal regularity, in Sobolev spaces, of the solution of a parabolic equation set in a triangular domain T. The right-hand term of the equation is taken in Lebesgue space L^p(T). The method of operators sums in the non-commutative case is referred to.

Sur une équation parabolique dans un domaine non cylindrique

We give some results about the optimal regularity of a solution to a parabolic equation, set in non cylindrical domains U=⋃t∈]0,1[{t}× It with It={x:0 1/2 is considered and the optimal regularity is obtained when the second member is regular. We use Labbas and Terrenis results [7]. This study is generalized when ϕϕ′ is Holderian. The second model corresponds to the limit case ϕ(t)=t and the maximal regularity is obtained for second members taken only in Lp(U), 1<p<∞. Here, we use Dore–Vennis results [3]. To cite this article: R. Labbas et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 1017–1022.

Study of boundary value and transmission problems in the Hölder spaces

Abstract We present new results on the resolution of singular transmission problems in Holder spaces completing in this way the work in L p cases given in [A. Favini, R. Labbas, K. Lemrabet, S. Maingot, Study of the limit of transmission problems in a thin layer by the sum theory of linear operators, Rev. Mater. Comput. 18 (1) (2005) 143–176]. Our approach makes use of the impedance notion operator [M.A. Leontovich, Approximate boundary conditions for the electromagnetic field on the surface of a good conductor. Investigations on radiowave propagation, Moscow Acad. Sci. (Part II) (1948)] which leads to obtain direct and simplified problems. We then use the Dunford calculus and some similar techniques as in [R. Labbas, Problemes aux limites pour une equation differentielle abstraite de type elliptique, These d’etat, Universite de Nice, 1987; A. El Haial, R. Labbas, On the ellipticity and solvability of abstract second-order differential equation, Electron. J. Differ. Eq. 57 (2001) 1–18; A. Favini, R. Labbas, S. Maingot, H. Tanabe, A. Yagi, Unified study of elliptic problems in Holder spaces, C. R. Acad. Sci. Paris Ser. 1341 (2005)] in order to prove existence, uniqueness and maximal regularities results.

Abstract differential equations of elliptic type with general Robin boundary conditions in Hölder spaces

In this article, we prove some new results on abstract second-order differential equations of elliptic type with general Robin boundary conditions. The study is performed in Hölder spaces and uses the well-known Da Prato–Grisvard sum theory. We give necessary and sufficient conditions on the data to obtain a unique strict solution satisfying the maximal regularity property. This work completes the ones studied by Favini et al. [A. Favini, R. Labbas, S. Maingot, H. Tanabe, and A. Yagi, Necessary and sufficient conditions in the study of maximal regularity of elliptic differential equations in Hölder spaces, Discrete Contin. Dyn. Syst. 22 (2008), pp. 973–987] and Cheggag et al. [M. Cheggag, A. Favini, R. Labbas, S. Maingot and A. Medeghri, Sturm-Liouville problems for an abstract differential equation of elliptic type in UMD spaces, Differ. Int. Eqns 21(9–10) (2008), pp. 981–1000].

MATRIX TRANSFORMATIONS AND GENERATORS OF ANALYTIC SEMIGROUPS

We establish a relation between the notion of an operator of an analytic semigroup and matrix transformations mapping from a set of sequences into, where is either of the sets,, or. We get extensions of some results given by Labbas and de Malafosse concerning applications of the sum of operators in the nondifferential case.