Stéphane Maingot

Study of complete abstract elliptic differential equations in non-commutative cases

This article is devoted to abstract second-order complete elliptic differential equations set on the whole line in a non-commutative framework. Existence, uniqueness and maximal regularity of the strict solution are proved. This study is performed in BUC θ(ℝ; X).

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].

Second order abstract differential equations of elliptic type set in R

Abstract In this paper we give some new results on complete abstract second order differential equations of elliptic type set in ℝ+. In the framework of UMD spaces, we use the celebrated Dore–Venni Theorem to prove existence and uniqueness for the strict solution. We will use also the Da Prato–Grisvard Sum Theory to furnish results when the space is not supposed to be a UMD space.

Singularities in boundary value problems for an abstracts second order differential equation of elliptic type

In this work we give an alternative approach to the study of some singular boundary value problems for a second order differential-operator equation in the space of Holder continuous functions. We prove that the solution can be represented explicitly as the sum u = ur + us °f a regular part and a singular part under some natural assumptions on the data. We then give a complete analysis of ur and ug by using the operational calculus.