Capacity Solution to a Nonlinear Elliptic Coupled System in Orlicz–Sobolev Spaces

2020 ◽  
Vol 17 (2) ◽  
Author(s):  
H. Moussa ◽  
F. Ortegón Gallego ◽  
M. Rhoudaf
Keyword(s):  
Sobolev Spaces ◽  
Coupled System ◽  
Nonlinear Elliptic

Existence Results for Nonlinear Elliptic Equations with Leray–Lions Operators in Sobolev Spaces with Variable Exponents

2021 ◽  
Vol 110 (5-6) ◽  
pp. 830-841
Author(s):  
M. Mosa Al-Shomrani ◽  
M. Ben Mohamed Salah ◽  
A. Ghanmi ◽  
K. Kefi
Keyword(s):  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Existence Results ◽  
Variable Exponents ◽  
Nonlinear Elliptic

scholarly journals Existence result for a nonlinear elliptic problem by topological degree in Sobolev spaces with variable exponent

2021 ◽  
Vol 7 (1) ◽  
pp. 50-65
Author(s):  
Mustapha Ait Hammou ◽  
Elhoussine Azroul
Keyword(s):  
Sobolev Spaces ◽  
Elliptic Problem ◽  
Topological Degree ◽  
Variable Exponent ◽  
Existence Of Solutions ◽  
Nonlinear Elliptic Problem ◽  
Technical Approach ◽  
Nonlinear Elliptic ◽  
Variable Exponent Sobolev Spaces ◽  
Topological Degree Method

AbstractThe aim of this paper is to establish the existence of solutions for a nonlinear elliptic problem of the form\left\{ {\matrix{{A\left( u \right) = f} \hfill & {in} \hfill & \Omega \hfill \cr {u = 0} \hfill & {on} \hfill & {\partial \Omega } \hfill \cr } } \right.where A(u) = −diva(x, u, ∇u) is a Leray-Lions operator and f ∈ W−1,p′(.)(Ω) with p(x) ∈ (1, ∞). Our technical approach is based on topological degree method and the theory of variable exponent Sobolev spaces.

Solvability of strongly nonlinear elliptic variational problems in weighted Orlicz–Sobolev spaces

SeMA Journal ◽  
2019 ◽  
Vol 77 (2) ◽  
pp. 119-142 ◽  
Author(s):  
Abdelmoujib Benkirane ◽  
Mostafa El Moumni ◽  
Khaled Kouhaila
Keyword(s):  
Sobolev Spaces ◽  
Variational Problems ◽  
Strongly Nonlinear ◽  
Nonlinear Elliptic

The hydromechanical equilibrium state of poroelastic media with a static fracture: A dimension-reduced model with existence results in weighted Sobolev spaces and simulations with an XFEM discretization

2018 ◽  
Vol 28 (13) ◽  
pp. 2511-2556
Author(s):  
Katja K. Hanowski ◽  
Oliver Sander
Keyword(s):  
Weighted Sobolev Spaces ◽  
Coupled System ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Rock Matrix ◽  
Fracture Width ◽  
Nonlinear Elliptic ◽  
Static Fracture ◽  
Optimal Mesh ◽  
Lower Dimensional

We introduce a coupled system of partial differential equations for the modeling of the fluid–fluid and fluid–solid interactions in a poroelastic material with a single static fracture. The fluid flow in the fracture is modeled by a lower-dimensional Darcy equation, which interacts with the surrounding rock matrix and the fluid it contains. We explicitly allow the fracture to end within the domain, and the fracture width is an unknown of the problem. The resulting weak problem is nonlinear, elliptic and symmetric, and can be given the structure of a fixed-point problem. We show that the coupled fluid–fluid problem has a solution in a specially crafted Sobolev space, even though the fracture width cannot be bounded away from zero near the crack tip. For numerical simulations, we combine XFEM discretizations for the rock matrix deformation and pore pressure with a standard lower-dimensional finite element method for the fracture flow problem. The resulting coupled discrete system consists of linear subdomain problems coupled by nonlinear coupling conditions. We solve the coupled system with a substructuring solver and observe very fast convergence. We also observe optimal mesh dependence of the discretization errors even in the presence of crack tips.

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