A Local estimate for nonlinear equations with discontinuous coefficients

Kinnunen, Juha; Zhou, Shulin

doi:10.1080/03605309908821494

Cited by 184 publications

(130 citation statements)

References 15 publications

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“…It improves the results of [11], [38], [4], [19] and extends the estimates of the gradient given in [46], [47] for solutions U ∈ W 1,p 0 (Ω). Estimates in Marcinkiewicz or Lorentz spaces are given in [44], [5].…”

Section: More Regularity Resultssupporting

confidence: 66%

On the Connection Between Two Quasilinear Elliptic Problems With Source Terms of Order 0 or 1

Hamid

Bidaut‐Véron

2010

Commun. Contemp. Math.

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We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of R N . The first one, of the forminvolves a source gradient term with natural growth, where β is nonnegative, λ > 0, f (x) ≧ 0, and α is a nonnegative measure. The second one, of the formpresents a source term of order 0, where g is nondecreasing, and µ is a nonnegative measure. Here β and g can present an asymptote. The correlation gives new results of existence, nonexistence, regularity and multiplicity of the solutions for the two problems, without or with measures. New informations on the extremal solutions are given when g is superlinear.

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Section: More Regularity Resultssupporting

confidence: 66%

On the Connection Between Two Quasilinear Elliptic Problems With Source Terms of Order 0 or 1

Hamid

Bidaut‐Véron

2010

Commun. Contemp. Math.

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“…the case of systems, was treated by DiBenedetto and Manfredi [18]. For similar results concerning equations with VMO-coefficients we refer to [31], [32] and the references therein. Finally, a complete Calderón-Zygmund theory for equations with non-standard p(x)-growth was achieved by Acerbi and Mingione [3].…”

Section: Introductionmentioning

confidence: 76%

Higher integrability for parabolic systems with non-standard growth and degenerate diffusions

Bögelein¹,

Duzaar²

2011

Publicacions Matemàtiques

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The aim of this paper is to establish a Meyer's type higher integrability result for weak solutions of possibly degenerate parabolic systems of the typeThe vector-field a is assumed to fulfill a non-standard p(x, t)-growth condition. In particular it is shown that there exists ε > 0 depending only on the structural data such that there holds:loc , together with a local estimate for the p(·)(1 + ε)-energy.

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“…There are numerous papers devoted to the L q bound (1.6) for solutions of (1.1) in the super-natural range q > p. The pioneer work [19] dealt with the case Ω = R n , and the paper [21] obtained a local interior bound. In [6], using a perturbation technique developed for fully nonlinear PDEs [5], the authors proved certain local W 1,q regularity for an associated homogeneous quasilinear equation.…”

Section: Definition 11 (Uniform P-thickness)mentioning

confidence: 99%

Global Lorentz and Lorentz–Morrey estimates below the natural exponent for quasilinear equations

Adimurthi

Phuc

2015

Calc. Var.

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Abstract. Lorentz and Lorentz-Morrey estimates are obtained for gradients of very weak solutions to quasilinear equations of the formwhere div A(x, ∇u) is modelled after the p-Laplacian, p > 1. The estimates are global over bounded domains that satisfy a mild exterior uniform thickness condition that involves the p-capacity. The vector field datum f is allowed to have low degrees of integrability and thus solutions may not have finite L p energy. A higher integrability result at the boundary of the ground domain is also obtained for infinite energy solutions to the associated homogeneous equations.

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A Local estimate for nonlinear equations with discontinuous coefficients

Cited by 184 publications

References 15 publications

On the Connection Between Two Quasilinear Elliptic Problems With Source Terms of Order 0 or 1

On the Connection Between Two Quasilinear Elliptic Problems With Source Terms of Order 0 or 1

Higher integrability for parabolic systems with non-standard growth and degenerate diffusions

Global Lorentz and Lorentz–Morrey estimates below the natural exponent for quasilinear equations

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