Approximated solutions of equations withL
1 data. Application to theH-convergence of quasi-linear parabolic equations

Dall’Aglio, Andrea

doi:10.1007/bf01758989

Cited by 161 publications

(54 citation statements)

References 8 publications

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“…Therefore, the notion of a solution has to be weakened and in the literature there have been introduced different approaches. We shall follow the one in [9,10,14] (Ω) associated with the problem. On the other hand, they do belong to W 1,p(·)−1…”

Section: And There Exists a Constantmentioning

confidence: 99%

“…In the literature there are different approaches, see [8,40,49]. We shall follow the one introduced in [9,10,14] (Ω) and therefore the weak formulation of (1.11) makes sense for a SOLA. Moreover, in the case that µ ∈ L 1 (Ω) the SOLA is unique, in the class of such solutions.…”

Section: Gradient Estimate For General Solutionsmentioning

confidence: 99%

“…(Ω)-of a SOLA in the case of coefficients with non standard growth in the spirit of [9,10,14] via the methods of [8,40,49]. For the sake of briefness we shall not give the whole proof here, but only the main steps and the approximation procedure leading to the gradient estimate (1.10) for a SOLA.…”

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confidence: 99%

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Gradient estimates via non standard potentials and continuity

Bögelein

Habermann

2010

Ann. Acad. Sci. Fenn. Math.

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Abstract. We consider elliptic problems with non standard growth conditions whose most prominent model example is the p(x)-Laplacean equationwith a measure data right-hand side µ. We prove pointwise gradient estimates in terms of a non standard version of the non-linear Wolff potential of the right-hand side measure, and moreover a characterization for C 1 -regularity of the solution, also in terms of the Wolff potential. The C 1 -regularity criterion is also related to the density of µ and the decay rate of its L n -norm on small balls. Moreover, from the pointwise gradient estimates the Calderón and Zygmund theory and several types of local estimates follow as a consequence.

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Section: And There Exists a Constantmentioning

confidence: 99%

Section: Gradient Estimate For General Solutionsmentioning

confidence: 99%

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Gradient estimates via non standard potentials and continuity

Bögelein

Habermann

2010

Ann. Acad. Sci. Fenn. Math.

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“…(9) θ is a renormalized solution of (2)-(3). (10) Under regularities (7)- (8), the right-hand side of (2), (A(x)Du − f (θ)) · Du, belongs to L 1 (θ). So we are in the framework of renormalized solution for equation (2).…”

Section: Assumptions and Definitionsmentioning

confidence: 99%

“…Renormalized solutions have been introduced by R.J. DiPerna and P.-L. Lions in [11] and [12] for first order equations and have been adapted for elliptic equations in [5], [18], [19] and for elliptic equations with general measure data in [9] (see also [8]). Other frameworks as entropy solutions [2] or SOLA [10] may be used for equation (2) with L 1 data.…”

Section: Introductionmentioning

confidence: 99%