Leray-Lions Operator Research Articles

Entropy solutions for some nonlinear and noncoercive parabolic problems in the anisotropic Sobolev spaces

In this paper, we studied the existence of solutions for a class parabolic problem involving a non-coercive Leray-Lions operator with a Hardy potential. Using an approximate method and some a priori estimation techniques, we show the existence of entropy solutions for the studied problem within the framework of anisotropic Sobolev spaces.

Existence results for the time-incremental elastic contact problem with Coulomb friction in 2D

In this paper, the structure of the incremental quasistatic contact problem with Coulomb friction in linear elasticity (Signorini–Coulomb problem) is unraveled and sharp existence results are proved for the most general two-dimensional problem with arbitrary geometry and elasticity modulus tensor. The problem is reduced to a variational inequality involving a nonlinear operator which handles both elasticity and friction. This operator is proved to fall into the class of the so-called Leray–Lions operators, so that a result of Brézis can be invoked to solve the variational inequality. It turns out that one property in the definition of Leray–Lions operators is difficult to check and requires proving a new fine property of the linear elastic Neumann-to-Dirichlet operator. This fine property is only established in the case of the two-dimensional problem, limiting currently our existence result to that case. In the case of isotropic elasticity, either homogeneous or heterogeneous, the existence of solutions to the Signorini–Coulomb problem is proved for arbitrarily large friction coefficient. In the case of anisotropic elasticity, an example of nonexistence of a solution for large friction coefficient is exhibited and the existence of solutions is proved under an optimal condition for the friction coefficient.

Anisotropic obstacle Neumann problems in weighted Sobolev spaces and variable exponent

Abstract In this paper we focus on a certain class of anisotropic obstacle problems governed by a Leray–Lions operator. This problem is subject to homogeneous Neumann boundary conditions. By applying truncation techniques and the monotonicity method, we establish the existence of entropy solutions for the problem studied in the framework of anisotropic weighted Sobolev spaces with variable exponent.

Existence and multiplicity of weak solutions for a system of fourth‐order elliptic equations with combined nonlocal and indefinite source terms

This manuscript aims to investigate the existence of weak solutions for a system of partial differential equations (PDEs) described by Equation (1.2). The system consists of a set of PDEs with Leray–Lions operators and nonlinear terms. The goal is to establish the existence of at least one nontrivial weak solution and at least three weak solutions for the system. The PDEs are defined on a bounded domain in , with , and subject to appropriate boundary conditions. The system involves various parameters, functions, and growth conditions that are carefully defined throughout the paper. The study focuses on understanding the behavior and existence of solutions for this system of PDEs, which has applications in physics, engineering, and other scientific fields.

Three weak solutions for a class of fourth order p(x)-Kirchhoff type problem with Leray-Lions operators

In this work, we study the multiplicity of a weak solution for a fourth order p(x)-Kirchhoff type problem involving the Leray-Lions type operators with no flux boundary condition. By using variational approach and critical point theory, we determine an open interval of parameters for which our problem admits at least three distinct weak solutions.

Increase of power leads to a bilateral solution to a strongly nonlinear elliptic coupled system

Abstract In this paper, we analyze the following nonlinear elliptic problem A ( u ) = ρ ( u ) | ∇ φ | 2 in Ω , div ( ρ ( u ) ∇ φ ) = 0 in Ω , u = 0 on ∂ Ω , φ = φ 0 on ∂ Ω . $\begin{cases}A\left(u\right)=\rho \left(u\right)\vert \nabla \varphi {\vert }^{2}\,\text{in}\,{\Omega},\quad \hfill \\ \text{div}\left(\rho \left(u\right)\nabla \varphi \right)=0\,\text{in}\,{\Omega},\quad \hfill \\ u=0\,\text{on}\,\partial {\Omega},\quad \hfill \\ \varphi ={\varphi }_{0}\,\text{on}\,\partial {\Omega}.\quad \hfill \end{cases}$ where A(u) = −div a(x, u, ∇u) is a Leray-Lions operator of order p. The second member of the first equation is only in L 1(Ω). We prove the existence of a bilateral solution by an approximation procedure, the keypoint being a penalization technique.

Non-coercive nonlinear parabolic problem with measurable obstacle in modular space

In this paper, we shall be concerned with the existence result for an unilateral problem associated to the form, ∂u ∂t −div ( a(x, t, u, ∇u) ) −div(Φ(x, t, u))) = f in QT = Ω× (0, T ), where Au = −diva(x, t, u, ∇u) is a Leray-Lions operator defined on W 1,x 0 LM (QT ), The growth and coercivity conditions on the monotone vector field a are prescribed by generalized N -function M , which does not have to satisfy the ∆2 condition. Therefore we work with a generalized Musielak-Orlicz spaces which are not necessarily reflexive. The right hand side and the initial data belong to L1(QT ) and the lower order term Φ is a non-coercive Carath ́eodory function.

Existence and Uniqueness of Renormalized Solution to Nonlinear Anisotropic Elliptic Problems with Variable Exponent and L 1 -Data

Nonlinear partial differential equations are considered as an essential tool for describing the behavior of many natural phenomena. The modeling of some phenomena requires to work in Sobolev spaces with constant exponent. But for others, such as electrorheological fluids, the properties of classical spaces are not sufficient to have precision. To overcome this difficulty, we work in the appropriate spaces called Lebesgue and Sobolev spaces with variable exponent. In recent works, researchers are attracted by the study of mathematical problems in the context of variable exponent. This great interest is motivated by their applications in many fields such as elastic mechanics, fluid dynamics, and image restoration. In this paper, we combine the technic of monotone operators in Banach spaces and approximation methods to prove the existence of renormalized solutions of a class of nonlinear anisotropic problem involving p ⟶ . − Leray–Lions operator, a graph, and L 1 data. In particular, we establish the uniqueness of the solution when the graph data are considered a strictly increasing function.

Existence of three weak solutions for fourth-order Leray–Lions problem with indefinite weights

In this paper, we establish the existence of at least three weak solutions for the fourth-order problem with indefinite weights involving the Leray–Lions operator with nonstandard growth conditions. We generalize the result obtained by Kefi et al. [On the fourth-order Leray–Lions problem with indefinite weight and nonstandard growth conditions. Bull Math Sci. 2021;DOI:10.1142/S1664360721500089]. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano [On the structure of the critical set of nondifferentiable functions with a weak compactness condition. Appl Anal. 2010;89:1–10].

Nonlinear elliptic equations with measure data in Orlicz spaces

UDC 517.5 In this article, we study the existence result of the unilateral problemwhere is a Leray–Lions operator defined on Sobolev–Orlicz space where and are two complementary -functions, the first and the second lower terms and satisfies only the growth condition and any sign condition is assumed and where is a measurable function.

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

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

On the fourth-order Leray–Lions problem with indefinite weight and nonstandard growth conditions

We prove the existence of at least three weak solutions for the fourth-order problem with indefinite weight involving the Leray–Lions operator with nonstandard growth conditions. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano [Appl. Anal. 89 (2010) 1–10].

Design and convergence analysis of numerical methods for stochastic evolution equations with Leray–Lions operator

Abstract The gradient discretization method (GDM) is a generic framework, covering many classical methods (finite elements, finite volumes, discontinuous Galerkin, etc.), for designing and analysing numerical schemes for diffusion models. In this paper we study the GDM for a general stochastic evolution problem based on a Leray–Lions type operator. The problem contains the stochastic $p$-Laplace equation as a particular case. The convergence of the gradient scheme (GS) solutions is proved by using discrete functional analysis techniques, Skorohod theorem and the Kolmogorov test. In particular, we provide an independent proof of the existence of weak martingale solutions for the problem. In this way we lay foundations and provide techniques for proving convergence of the GS approximating stochastic partial differential equations.

Infinitely many solutions for a generalized p(·)‐Laplace equation involving Leray–Lions type operators

We study the existence of infinitely many solutions for a generalized p(·)‐Laplace equation involving Leray–Lions operators. Firstly, under a p(·)‐sublinear condition for nonlinear term, we obtain a sequence of solutions approaching 0 by showing a new a priori bound for solutions. Secondly, for a p(·)‐superlinear condition, we produce a sequence of solutions whose Sobolev norms diverge to infinity when the nonlinear term satisfies a couple of generalized Ambrosetti–Rabinowitz type conditions in which each associated energy functional holds the Palais–Smale condition. Lastly, we deal with a case without the Ambrosetti–Rabinowitz type condition in which an associated energy functional holds the Cerami condition and establish a sequence of solutions whose Sobolev norms diverge to infinity.

Existence of <i>T</i>-<i>ν</i>-<i>p</i>(<i>x</i>)-Solution of a Nonhomogeneous Elliptic Problem with Right Hand Side Measure

Using the theory of weighted Sobolev spaces with variable exponent and the L1-version on Minty’s lemma, we investigate the existence of solutions for some nonhomogeneous Dirichlet problems generated by the Leray-Lions operator of divergence form, with right-hand side measure. Among the interest of this article is the given of a very important approach to ensure the existence of a weak solution of this type of problem and of generalization to a system with the minimum of conditions.

Existence of weak solutions for a nonlinear parabolic equations by Topological degree

We study the nonlinear parabolic initial boundary value problem associated to the equation ut − diva(x, t, u, grad u) = f(x, t), where the terme − diva(x, t, u, grad u) is a Leray-Lions operator, The right-hand side f is assumed to belong to L^q(Q). We prove the existence of a weak solution for this problem by using the Topological degree theory for operators of the form L + S, where L is a linear densely defined maximal monotone map and S is a bounded demicontinuous map of class (S+) with respect to the domain of L.

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

Abstract The aim of this paper is to establish the existence of solutions for a nonlinear elliptic problem of the form { A ( u ) = f i n Ω u = 0 o n ∂ Ω \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.

Some class of nonlinear inequalities with gradient constraints in Orlicz spaces

Abstract In the present paper, we show the existence of solutions of some nonlinear inequalities of the form 〈Au + g(x, u,∇ u), v −u〉 ≥〈 f, v −u〉 with gradient constraint that depend on the solution itself, where A is a Leray-Lions operator defined on Orlicz spaces, g is a nonlinearity and f ∈ L 1.

Existence results for a perturbed Dirichlet problem without sign condition in Orlicz spaces

UDC 517.5 We deal with the existence result for nonlinear elliptic equations related to the form < b r > A u + g ( x , u , ∇ u ) = f , < b r > where the term - ⅆ i v ( a ( x , u , ∇ u ) ) is a Leray–Lions operator from a subset of W 0 1 L M ( Ω ) into its dual. The growth and coercivity conditions on the monotone vector field a are prescribed by an N -function M which does not have to satisfy a Δ 2 -condition. Therefore we use Orlicz–Sobolev spaces which are not necessarily reflexive and assume that the nonlinearity g ( x , u , ∇ u ) is a Carathéodory function satisfying only a growth condition with no sign condition. The right-hand side~ f belongs to W -1 E M ¯ ( Ω ) .

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

We analyze the existence of a capacity solution to the following nonlinear elliptic coupled system, whose unknowns are the temperature inside a semiconductor material, u, and the electric potential, $$\varphi $$, $$\begin{aligned} \left\{ \begin{array}{ll} -Au= \rho (u)|\nabla \varphi |^2 &{}\quad \mathrm{in}\ \Omega , \\ \mathop {\mathrm{div}}\nolimits (\rho (u)\nabla \varphi ) =0&{}\quad \mathrm{in}\ \Omega , \\ \varphi =\varphi _0 &{}\quad \hbox {on } \partial \Omega , \\ u=0&{}\quad \mathrm{on}\, {\partial \Omega }, \end{array} \right. \end{aligned}$$where $$\Omega \subset {\mathbb {R}}^d$$, $$d\ge 2$$ and $$\displaystyle Au=-\mathop {\mathrm{div}}\nolimits a(x,u,\nabla u)$$ is a Leray–Lions operator defined on $$W_0^{1}L_M(\Omega )$$, M is a N-function which does not have to satisfy a $$\Delta _2$$ condition. Therefore, we work with generalized Orlicz–Sobolev spaces which are not necessarily reflexive. The function $$\varphi _0$$ is given. The proof combines truncation methods, monotonicity techniques and regularizing methods in Orlicz spaces. We introduce a sequence of approximate problems which converges (up to a subsequence) in a certain sense to a capacity solution in the context of non-reflexive Orlicz–Sobolev spaces.