Existence of bounded solutions for non linear elliptic unilateral problems

Boccardo, Lucio; Murat, François; Puel, Jean-Pierre

doi:10.1007/bf01766148

Cited by 245 publications

(119 citation statements)

References 8 publications

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“…Moreover, as in the proof of Theorem 2.1 and [11, Theorem 5.1], relying on the strict monotonicity of a and utilizing an argument of [24,21], we also deduce the strong convergence of ∇ T w T,∆t to ∇w in L p (Q).…”

Section: Convergence and Statement Of Main Resultsmentioning

confidence: 59%

“…Starting off from (14), we can use the Minty-Browder trick (see, for example, [64,24,4,21] and the proof of Theorem 7.1 in Section 7) to deduce that…”

Section: Definition 22 (Entropy Solution) An Entropy Solution Of Thmentioning

confidence: 99%

“…[24], [21,Lemma 5], [42,Lemma 8.4]. Due to the coercivity of a(·), | ∇w ρ | p ρ is equi-integrable, so the Vitali theorem yields the strong L p convergence of ∇w ρ , along a subsequence if necessary, to a limit already identified as ∇w, w = A(u).…”

Section: Definition 22 (Entropy Solution) An Entropy Solution Of Thmentioning

confidence: 99%

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Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

Andreïanov¹,

Boyer²,

Hubert³

2006

Numerical Methods Partial

128

275

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Abstract. We consider a class of doubly nonlinear degenerate hyperbolicparabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès [41]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around some basic a priori estimates, the discrete duality features, Minty-Browder type arguments, and "hyperbolic" L ∞ weak-⋆ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna, . . . ). Our results cover the case of non-Lipschitz nonlinearities.

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Section: Convergence and Statement Of Main Resultsmentioning

confidence: 59%

“…Starting off from (14), we can use the Minty-Browder trick (see, for example, [64,24,4,21] and the proof of Theorem 7.1 in Section 7) to deduce that…”

Section: Definition 22 (Entropy Solution) An Entropy Solution Of Thmentioning

confidence: 99%

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Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

Andreïanov¹,

Boyer²,

Hubert³

2006

Numerical Methods Partial

128

275

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“…The following lemma generalizes to the weighted case the analogous Lemma 5 in [7]. For that, we use the method of [7] and [13] which gives the strong convergence of u n .…”

Section: Some Technical Lemmasmentioning

confidence: 96%

Existence Results for Quasilinear Degenerated Equations Via Strong Convergence of Truncations

Akdim¹,

Azroul²,

Benkirane³

2004

Rev. Mat. Complut.

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“…Remark that this condition corresponds, in the classical Sobolev spaces W 1,p to the case p < N, which is the interesting case in these spaces. Further works for the unilateral problem corresponding to (1.1) in the L p case can be found in [13,14,15].…”

Section: Ait Khellou and A Benkiranementioning

confidence: 99%