Browsing by Author "Supervisor: Abdelaziz, Hellal"

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    On semilinear elliptic equations involving stummel classes
    (Mohamed Boudiaf University of M'sila, 2025-06-15) Abdelkrim, Maghni; Supervisor: Abdelaziz, Hellal; Co-Supervisor: Noureddine, Dechoucha
    This memory deals with the study of the existence, uniqueness, and regularity of semilinear elliptic equations, more precisely: ( −d i v(M(x)∇v)+ g (v) = f , v ∈ W 1,2 0 (Ω). Where Ω is a bounded open subset of R n (n Ê 3), In this case, we suppose that M(x) is nxn symmetric matrix, elliptic, bounded, and g : R → R is non decreasing, and Lipschitz. The datum f is taken belongs to S˜α(Ω), for α = 1or 2. Since the Stummel classes have some inclusion properties with other function spaces and appli cations to the regularity of the solution of elliptic partial differential equations, so our studying is based on employing Stampacchia’s lemma and a weighted embedding of a function in Stummel classes where the weight is in compactly supported Sobolev spaces.
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    On Some Elliptic Equations With W0 1,1(Ω) Solutions
    (Mohamed Boudiaf University of M'sila, 2025-06-15) Djihane, Hamidi; Supervisor: Abdelaziz, Hellal
    This work investigates the regularizing effects of lower-order terms in nonlinear Dirichlet prob lems of the form: ( − u div = 0, on ¡ |∇u| p−2∇u ¢ + H(x,u,∇u) = f (x), in Ω ∂ , Ω, (1) where Ω ⊂ R N (N ≥ 2) is a bounded domain, 1 < p ≤ N, and f has poor summability. We demon strate how lower-order terms can enhance solution regularity, particularly when f ∈ L 1 (Ω) or other Lebesgue spaces. According to the work [8], this study focuses on four principal cases: (A) For H(x,u,∇u) = u|u| r−2 , we establish existence of weak solutions in W0 1,2(Ω) even when f ∈ L 1 (Ω) (B) With polynomial nonlinearities, we prove existence of distributional solutions in W0 1,1(Ω) for f ∈ L r ′ /p (Ω) (1 < p ≤ r ′ ) (C) For gradient-dependent terms H(x,u,∇u) = u|u| r−2 |∇u|, we obtain solutions in W0 1,1(Ω) ∩ L r−1 (Ω) when f ∈ L 1 (Ω) and 1 < r ≤ N N (p − − 1 1) (D) We compare these results with the semilinear case (p = 2), highlighting differences in regu larization mechanisms The analysis employs a unified three-step approach: (1) approximation by regular problems, (2) derivation of a priori estimates in W0 1,1(Ω), and (3) passage to the limit. Our results significantly ex tend previous work by demonstrating existence in borderline cases where the unperturbed prob lem (H = 0) admits no solutions. The findings have important implications for understanding nonlinear elliptic equations with non-regular data.
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    Some Properties of Musielak-Orlicz-Sobolev spaces with an application in Nonlinear PDE
    (Mohamed Boudiaf University of M'sila, 2025-06-15) Riane Ade, Loumi; Supervisor: Abdelaziz, Hellal
    This work investigates some properties of Musielak-Orlicz-Sobolev spaces W1,θ(Ω) and their application to the nonlinear double-phase problem with variable exponents:    −div |∇u| p(x)−2∇u + µ(x)|∇u| q(x)−2∇u = f(x, u,∇u), in Ω, u = 0, on ∂Ω, (4.39) Where Ω ⊂ RN (N ≥ 2) is a bounded domain with Lipschitz boundary ∂Ω, p, q ∈ C(Ω) satisfy 1 < p(x) < N, p(x) < q(x) for all x ∈ Ω, µ ∈ L ∞(Ω), and f is a Carathéodory function. We study the existence and uniqueness results for solutions to problem (4.39) in the Musielak Orlicz-Sobolev space framework. The variational methods cannot be applied here due to problem (4.39) does not have variational structure. This why our approach employs non variational method following [20], building on the analytical techniques developed in [12] and [14] for studying partial differential equations with non-standard growth conditions. These methods provide a robust framework for analyzing sequences of approximate solu tions and their convergence properties. The analysis highlights the fundamental role of Musielak-Orlicz-Sobolev spaces in functional analysis, extending the classical variable exponent Lebesgue space theory [3]. We examine several key properties of these spaces that are essential for handling the nonlinearities and variable growth conditions in problem (4.39).