Topological degree methods for partial differential operators in generalized Sobolev spaces

2021 ◽  
Vol 39 (2) ◽  
pp. 39-61
Author(s):  
Mustapha Ait Hammou ◽  
Elhoussine Azroul ◽  
Badr Lahmi
Keyword(s):  
Topological Degree ◽  
Nonlinear Elliptic Equation ◽  
Differential Operators ◽  
Existence Of Solutions ◽  
Divergence Form ◽  
Generalized Sobolev Spaces ◽  
Nonlinear Elliptic ◽  
Partial Differential Operators ◽  
Topological Degree Methods ◽  
General Divergence

The main aim of this paper is to prove, by using the topological degree methods, the existence of solutions for nonlinear elliptic equation Au = f where Au  is partial dierential operators of general divergence form.

scholarly journals Topological degree methods for a Neumann problem governed by nonlinear elliptic equation

2020 ◽  
Vol 6 (2) ◽  
pp. 231-242
Author(s):  
Adil Abbassi ◽  
Chakir Allalou ◽  
Abderrazak Kassidi
Keyword(s):  
Elliptic Equation ◽  
Topological Degree ◽  
Nonlinear Elliptic Equation ◽  
Neumann Boundary ◽  
Existence Of Weak Solutions ◽  
Bounded Smooth Domain ◽  
Nonlinear Elliptic ◽  
Topological Degree Methods ◽  
Bounded Smooth ◽  
Neumann Boundary Value Problems

AbstractIn this paper, we will use the topological degree, introduced by Berkovits, to prove existence of weak solutions to a Neumann boundary value problems for the following nonlinear elliptic equation- div\,\,a\left( {x,u,\nabla u} \right) = b\left( x \right){\left| u \right|^{p - 2}}u + \lambda H\left( {x,u,\nabla u} \right),where Ω is a bounded smooth domain of 𝕉N.

scholarly journals The Dirichlet Problem for Second-Order Divergence Form Elliptic Operators with Variable Coefficients: The Simple Layer Potential Ansatz

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Alberto Cialdea ◽  
Vita Leonessa ◽  
Angelica Malaspina
Keyword(s):  
Dirichlet Problem ◽  
Differential Operators ◽  
Differential Forms ◽  
Variable Coefficients ◽  
Divergence Form ◽  
Boundary Integral ◽  
Second Order ◽  
Boundary Integral Method ◽  
Layer Potential ◽  
Partial Differential Operators

We investigate the Dirichlet problem related to linear elliptic second-order partial differential operators with smooth coefficients in divergence form in bounded connected domains ofRm(m≥3) with Lyapunov boundary. In particular, we show how to represent the solution in terms of a simple layer potential. We use an indirect boundary integral method hinging on the theory of reducible operators and the theory of differential forms.

Perturbation of nonlinear partial differential variational inequalities, II

1978 ◽  
Vol 82 (1-2) ◽  
pp. 153-170
Author(s):  
Elena Stroescu
Keyword(s):  
Variational Inequality ◽  
Differential Operators ◽  
Cartesian Product ◽  
Divergence Form ◽  
Partial Differential ◽  
Convergence Of Solutions ◽  
Direct Sum ◽  
Partial Differential Operators ◽  
Compactness Theorems ◽  
Differential Variational Inequalities

SynopsisThis paper is devoted to the study of the weak respectively strong convergence of solutions of a variational inequality, with nonlinear partial differential operators of the generalized divergence form and of semimonotone type, under a perturbation of the domain of definition. In this study we use abstract convergence theorems given by Stroescu and Vivaldi, convergence concepts defined according to Stummel and compactness theorems of the natural imbedding of the Cartesian product of Sobolev spaces into the direct sum of Lp spaces, also by Stummel.

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.

A pointwise differential inequality and second-order regularity for nonlinear elliptic systems

2021 ◽  
Author(s):  
Anna Kh. Balci ◽  
Andrea Cianchi ◽  
Lars Diening ◽  
Vladimir Maz’ya
Keyword(s):  
Differential Inequality ◽  
Differential Operators ◽  
Elliptic Systems ◽  
Second Order ◽  
Nonlinear Elliptic Systems ◽  
Nonlinear Elliptic ◽  
Regularity Properties ◽  
Partial Differential Operators ◽  
Global Estimates ◽  
Special Case

AbstractA sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic systems in domains in $${\mathbb R^n}$$ R n are derived. Both local and global estimates are established. Minimal assumptions on the boundary of the domain are required for the latter. In the special case of the p-Laplace system, our conclusions broaden the range of the admissible values of the exponent p previously known.

Positive solutions for the Robin p-Laplacian problem with competing nonlinearities

2019 ◽  
Vol 12 (1) ◽  
pp. 31-56 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Nonlinear Elliptic Equation ◽  
Differential Operators ◽  
Comparison Principles ◽  
Reaction Term ◽  
Nonlinear Elliptic ◽  
Carathéodory Function ◽  
Concave And Convex Terms

AbstractWe consider a parametric nonlinear elliptic equation driven by the Robin p-Laplacian. The reaction term is a Carathéodory function which exhibits competing nonlinearities (concave and convex terms). We prove two bifurcation-type results describing the set of positive solutions as the parameter varies. In the process we also prove two strong comparison principles for Robin equations. These results are proved for differential operators which are more general than the p-Laplacian and need not be homogeneous.

scholarly journals Existence of solutions for elliptic equations having natural growth terms in Orlicz spaces

2004 ◽  
Vol 2004 (12) ◽  
pp. 1031-1045 ◽  
Author(s):  
A. Elmahi ◽  
D. Meskine
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Existence Result ◽  
Existence Of Solutions ◽  
Orlicz Spaces ◽  
Natural Growth ◽  
Strongly Nonlinear ◽  
Nonlinear Elliptic ◽  
Natural Growth Terms

Existence result for strongly nonlinear elliptic equation with a natural growth condition on the nonlinearity is proved.

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