Characterizations of the existence and removable singularities of divergence-measure vector fields

Phuc, Nguyen Cong; Torres, Mónica

doi:10.1512/iumj.2008.57.3312

Cited by 35 publications

(26 citation statements)

References 19 publications

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“…The following lemma from [37] involves the Morrey space M d (R d ) and is used in the proof of Theorem 1.7. We sketch the proof to emphasize its connection with the classical Boxing inequality:…”

Section: From Proposition 45 We Deduce the Poincaré-wirtinger Trace mentioning

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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Section: From Proposition 45 We Deduce the Poincaré-wirtinger Trace mentioning

confidence: 99%

A boxing Inequality for the fractional perimeter

Ponce¹,

Spector²

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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“…While this result is known (see Theorem 2.8 in [23]), given its importance, a complete and self-contained proof using Definition 2.5 of the q-capacity will be given.…”

Section: Lemma 213 Let E ⊂⊂ ω Be a Set Of Finite Perimeter In ω Andmentioning

confidence: 99%

On locally essentially bounded divergence measure fields and sets of locally finite perimeter

Comi

Payne

2017

Advances in Calculus of Variations

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Abstract. Chen, Torres and Ziemer ([9], 2009) proved the validity of generalized Gauss-Green formulas and obtained the existence of interior and exterior normal traces for essentially bounded divergence measure fields on sets of finite perimeter using an approximation theory through sets with a smooth boundary. However, it is known that the proof of a crucial approximation lemma contained a gap. Taking inspiration from a previous work of Chen and Torres ([7], 2005) and exploiting ideas of Vol'pert ([29], 1985) for essentially bounded fields with components of bounded variation, we present here a direct proof of generalized GaussGreen formulas for essentially bounded divergence measure fields on sets of finite perimeter which includes the existence and essential boundedness of the normal traces. Our approach appears to be simpler since it does not require any special approximation theory for the domains and it relies only on the Leibniz rule for divergence measure fields. This freedom allows one to localize the constructions and to derive more general statements in a natural way.

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“…Thus by a result in [15] (see estimate (3.1) in [15]) we have (note that by hypothesis q/(p − 1) > 1)…”

Section: Proof Of Theorem32mentioning

confidence: 77%

On Calderón-Zygmund theory for p- and $${\mathcal{A}}$$ -superharmonic functions

Phuc

2011

Calc. Var.

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Characterizations of the existence and removable singularities of divergence-measure vector fields

Cited by 35 publications

References 19 publications

A boxing Inequality for the fractional perimeter

A boxing Inequality for the fractional perimeter

On locally essentially bounded divergence measure fields and sets of locally finite perimeter

On Calderón-Zygmund theory for p- and $${\mathcal{A}}$$ -superharmonic functions

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