Uniqueness and maximal regularity for nonlinear elliptic systems of n--Laplace type with measure valued right hand side

Dolzmann, Georg; Hungerbühler, Norbert; Müller, Stefan

doi:10.1515/crll.2000.022

Cited by 72 publications

(84 citation statements)

References 15 publications

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“…As a consequence of our result, (see [18], [11]), we can obtain strong A ∞ -weights of the form w = e nu , where u is a distributional solution of −div(|∇u| n−2 ∇u) = µ whenever µ is a signed Radon measure with small total variation. Indeed, it follows from the results of [18] and [11] that every distributional solution u of the previous equation has the property that the weak L n -norm of the distributional gradient of u is controlled by (|µ|(R n )) 1 n−1 .…”

Section: B(a R)| B(ar)mentioning

confidence: 53%

Strong $A_{\infty}$-weights and scaling invariant Besov capacities

Costea¹

2007

Rev. Mat. Iberoam.

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This article studies strong A ∞ -weights and Besov capacities as well as their relationship to Hausdorff measures. It is shown that in the Euclidean space R n with n ≥ 2, whenever n − 1 < s ≤ n, a function u yields a strong A ∞ -weight of the form w = e nu if the distributional gradient ∇u has sufficiently small || · || L s,n−s (R n ; R n )-norm. Similarly, it is proved that if 2 ≤ n < p < ∞, then w = e nu is a strong A ∞ -weight whenever the BesovLower estimates of the Besov B p -capacities are obtained in terms of the Hausdorff content associated with gauge functions h satisfying the condition 1 0 h(t) p −1 dt t < ∞.

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Section: B(a R)| B(ar)mentioning

confidence: 53%

Strong $A_{\infty}$-weights and scaling invariant Besov capacities

Costea¹

2007

Rev. Mat. Iberoam.

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“…In the regularity theory of measure data problems, for a long while it has been an open issue to establish the validity of estimates in Marcinkiewicz spaces in the conformal case p = n. Specifically, for a SOLA solution u of (1.1) we have the inclusion Du ∈ M n (Ω) -see (6.10) below for the definitions. This has been first proved in [19] directly for the p-Laplacean system, see also [47] for different approaches. Here we would like to outline how estimates (1.…”

Section: Integrability Estimatesmentioning

confidence: 86%

Gradient estimates via non-linear potentials

Duzaar¹,

Mingione²

2011

ajm

220

188

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“…The following properties are well-known in case p < N, see [7], [25] and more delicate in case p = N, see [36] and [45], where they require more regularity on the domain, namely, R N \Ω is geometrically dense:…”

Section: Renormalized Solutionsmentioning

confidence: 99%

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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Uniqueness and maximal regularity for nonlinear elliptic systems of n--Laplace type with measure valued right hand side

Cited by 72 publications

References 15 publications

Strong $A_{\infty}$-weights and scaling invariant Besov capacities

Strong $A_{\infty}$-weights and scaling invariant Besov capacities

Gradient estimates via non-linear potentials

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

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