Browsing by Author "Supervisor: Hellal, Abdelaziz"

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    On Weak Lebesgue ( Marcinkiewicz ) Spaces with Variable Exponents
    (University of M'sila, 2023-06-10) Zerrouki, Fouzia; Supervisor: Hellal, Abdelaziz
    In this memory, we study the so-called Marcinkiewicz spaces ( weak Lebesgue spaces ) with variable exponents which are one of the first generalizations of the Lebesgue spaces. In the framework of Marcinkiewicz spaces we will study, among other topics, embedding results, convergence in measure, interpolation results, and the question of normability of the space. We also show a Fatou type lemma for Marcinkiewicz spaces as well as the completeness of the quasi-norm. The Lyapunov inequality and the H¨older inequality are shown to hold.
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    Some Existence and Regularity Results for Unilateral Problems with Degenerate Coercivity
    (University of M'sila, 2023-06-10) Naidji, Ali; Supervisor: Hellal, Abdelaziz
    In this work we study the existence and regularity of the solutions of unilateral problems 8>><>>: u ‚ 0, a.e.in ›, hAu,u ¡ vi • R› f (u ¡ v), 8v 2 H1 0(›),v ‚ 0, a.e.in ›, (1) Here › is a bounded, open subset of RN , with N ¨ 2, and Au ˘ ¡di v(a(x,u)Du) with a(x,s) : ›£R ! R is a Carathéodory function, satisfying the following conditions: fi (1¯jsj)θ • a(x,s) • fl. (2) for some real number θ such that. 0 • θ ˙ 1 (3) for almost every x 2 ›, for every s 2 R, where fi and fl are positive constants. The objective of this contribution is to study the existence and regularity of the solutions of unilateral problems (1) associated to A under assumptions (2)-(3) and with data f belonging to various Lebesgue space Lm(›) , for some m ¨ 1.