A limiting case for the divergence equation

Bousquet, Pierre; Mironescu, Petru; Russ, Emmanuel

doi:10.1007/s00209-012-1077-x

Cited by 14 publications

(15 citation statements)

References 19 publications

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“…ByḞ α,p q (Λ l R d ), we mean the space of differential l-forms on R d , the coefficients of which belong toḞ α,p q (R d ) (see Definition 2.1 below). The above statement extends the main result in [6] which was restricted to the conditions κ = 1 (which amounts to solving the equation div X = f with f ∈Ḟ α,p q ), α > 1/2 and p ≥ q ≥ 2; see also the earlier papers by Maz'ya [16] and also Mironescu [18] when κ = 1, and p = q = 2.…”

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confidence: 79%

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Approximation in higher-order Sobolev spaces and Hodge systems

Bousquet

Russ

Wang

et al. 2019

Journal of Functional Analysis

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Let d ≥ 2 be an integer, 1 ≤ l ≤ d − 1 and ϕ be a differential l-form on R d withẆ 1,d coefficients. It was proved by Bourgain and Brezis ([5, Theorem 5]) that there exists a differential l-form ψ on R d with coefficients in L ∞ ∩Ẇ 1,d such that dϕ = dψ. In the same work, Bourgain and Brezis also left as an open problem the extension of this result to the case of differential forms with coefficients in the higher order spaceẆ 2,d/2 or more generally in the fractional Sobolev spacesẆ s,p with sp = d. We give a positive answer to this question, provided that d − κ ≤ l ≤ d − 1, where κ is the largest positive integer such that κ < min(p, d). The proof relies on an approximation result (interesting in its own right) for functions inẆ s,p by functions inẆ s,p ∩L ∞ , even thoughẆ s,p does not embed into L ∞ in this critical case. The proofs rely on some techniques due to Bourgain and Brezis but the context of higher order and/or fractional Sobolev spaces creates various difficulties and requires new ideas and methods.

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confidence: 79%

“…Open problem 1.5. It is likely that Theorem 1.2 can be extended to the case of smooth bounded domains in R d , in the spirit of [6].…”

Section: Let Us End Up This Introduction With Three Open Problemsmentioning

confidence: 99%

Approximation in higher-order Sobolev spaces and Hodge systems

Bousquet

Russ

Wang

et al. 2019

Journal of Functional Analysis

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“…for nonnegative measures ν in R d which satisfy the ball-growth condition ν(B r (x)) ≤ r d−1 . This divergence equation is dual to inequality (1.6) for α = 1, and so we do not need to invoke any further results on special solutions to such an equation (see for example [10,13]). The latter case α → (k + 1) − is a direct consequence of lifting Theorem 1.3 via the mapping properties of Riesz potentials (see Lemma 4.6 below) and inherits the correct scaling from our Theorem 1.3.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A boxing Inequality for the fractional perimeter

Ponce¹,

Spector²

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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We prove the Boxing inequality:is the Hausdorff content of U of dimension d − α and the constant C > 0 depends only on d. We then show how this estimate implies a trace inequality in the fractional Sobolev space W α,1 (R d ) that includes Sobolev's L d d−α embedding, its Lorentz-space improvement, and Hardy's inequality. All these estimates are thus obtained with the appropriate asymptotics as α tends to 0 and 1, recovering in particular the classical inequalities of first order. Their counterparts in the full range α ∈ (0, d) are also investigated.

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“…For the Hölder case see also the recent review in Csató, Dacorogna, and Kneuss [11]. In addition, we also note that the non-uniqueness feature of the first order system (1) allows some existence results with more regularity than expected from the usual theorems, as the striking results of Bourgain and Brezis [9], coming from a non-linear selection principle of the solutions to a linear problem (see the extensions to the Dirichlet problem in Bousquet, Mironescu, and Russ [10]).…”

Section: Introductionmentioning

confidence: 61%