On Some Elliptic Equations With W0 1,1(Ω) Solutions
| dc.contributor.author | Djihane, Hamidi | |
| dc.contributor.author | Supervisor: Abdelaziz, Hellal | |
| dc.date.accessioned | 2025-07-07T08:43:22Z | |
| dc.date.available | 2025-07-07T08:43:22Z | |
| dc.date.issued | 2025-06-15 | |
| dc.description.abstract | 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. | |
| dc.identifier.uri | https://repository.univ-msila.dz/handle/123456789/46661 | |
| dc.language.iso | en | |
| dc.publisher | Mohamed Boudiaf University of M'sila | |
| dc.subject | Nonlinear Dirichlet problem | |
| dc.subject | Existence | |
| dc.subject | Regularity | |
| dc.subject | Regularizing effects | |
| dc.subject | Non-regular data | |
| dc.subject | W0 1 | |
| dc.subject | 1(Ω) | |
| dc.title | On Some Elliptic Equations With W0 1,1(Ω) Solutions | |
| dc.type | Thesis |