Existence and uniqueness of solutions of unilateral problems in Orlicz spaces

Aharouch, L.; Bennouna, Jaouad

doi:10.1016/j.na.2010.01.005

Cited by 24 publications

(15 citation statements)

References 8 publications

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“…The existence of a solution to (5.8) is obtained by a straightforward application of Theorem 1, p. 107 in [10]. Also we can easily check that the solution of (5.8) is unique [2] Now, we show that |∇u| ∈ L M (Q T ), and the estimates…”

Section: An Existence Resultsmentioning

confidence: 70%

“…Indeed, ρ(s) may converge to zero as |s| tends to infinity and as a result, if u is unbounded in Q T , the elliptic equation becomes degenerate at points where u is infinity and, therefore, no a priori estimates for ∇ϕ will be available and thus, ϕ may not belong to a Sobolev space. Instead of ϕ, we may consider the function Φ = ρ(u)|∇ϕ| 2 as a whole and then show that belongs to L 2 (Q T ) d . This means that a new formulation of the original system is possible and the solution to this new formulation will be called capacity solution.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Capacity solution to a coupled system of parabolic–elliptic equations in Orlicz–Sobolev spaces

Moussa

Gallego

Rhoudaf

2018

Nonlinear Differ. Equ. Appl.

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Abstract. The existence of a capacity solution to a coupled nonlinear parabolic-elliptic system is analyzed, the elliptic part in the parabolic equation being of the form − div a (x, t, u, ∇u). The growth and the coercivity conditions on the monotone vector field a are prescribed by an N -function, M , which does not have to satisfy a Δ2 condition. Therefore we work with Orlicz-Sobolev spaces which are not necessarily reflexive. We use Schauder's fixed point theorem to prove the existence of a weak solution to certain approximate problems. Then we show that some subsequence of approximate solutions converges in a certain sense to a capacity solution.Mathematics Subject Classification. 35K60, 35D05, 35J70, 46E30.

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Section: An Existence Resultsmentioning

confidence: 70%

Section: Introductionmentioning

confidence: 99%

Capacity solution to a coupled system of parabolic–elliptic equations in Orlicz–Sobolev spaces

Moussa

Gallego

Rhoudaf

2018

Nonlinear Differ. Equ. Appl.

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“…where f ∈ ðC 0 ðRÞÞ N and f ∈ L 1 (Ω). For more results, we refer the reader to [12][13][14][15][16][17][18][19][20][21][22][23][24] and references therein.…”

Section: Introductionmentioning

confidence: 99%

On some nonlinear anisotropic elliptic equations in anisotropic Orlicz space

Benslimane

Aberqi

Bennouna

2021

AJMS

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PurposeIn the present paper, the authors will discuss the solvability of a class of nonlinear anisotropic elliptic problems (P), with the presence of a lower-order term and a non-polynomial growth which does not satisfy any sign condition which is described by an N-uplet of N-functions satisfying the Δ2-condition, within the fulfilling of anisotropic Sobolev-Orlicz space. In addition, the resulting analysis requires the development of some new aspects of the theory in this field. The source term is merely integrable.Design/methodology/approachAn approximation procedure and some priori estimates are used to solve the problem.FindingsThe authors prove the existence of entropy solutions to unilateral problem in the framework of anisotropic Sobolev-Orlicz space with bounded domain. The resulting analysis requires the development of some new aspects of the theory in this field.Originality/valueTo the best of the authors’ knowledge, this is the first paper that investigates the existence of entropy solutions to unilateral problem in the framework of anisotropic Sobolev-Orlicz space with bounded domain.

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“…It is generally difficult to move essentially L p techniques to Orlicz spaces. For more work within this framework, we quote [10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

The Existence and Uniqueness of an Entropy Solution to Unilateral Orlicz Anisotropic Equations in an Unbounded Domain

Benslimane

Aberqi

Bennouna

2020

Axioms

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Existence and uniqueness of solutions of unilateral problems in Orlicz spaces

Cited by 24 publications

References 8 publications

Capacity solution to a coupled system of parabolic–elliptic equations in Orlicz–Sobolev spaces

Capacity solution to a coupled system of parabolic–elliptic equations in Orlicz–Sobolev spaces

On some nonlinear anisotropic elliptic equations in anisotropic Orlicz space

The Existence and Uniqueness of an Entropy Solution to Unilateral Orlicz Anisotropic Equations in an Unbounded Domain

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