Isoperimetric and Sobolev inequalities for Carnot-Carath�odory spaces and the existence of minimal surfaces

Garofalo, Nicola; Nhieu, Duy-Minh

doi:10.1002/(sici)1097-0312(199610)49:10<1081::aid-cpa3>3.0.co;2-a

Cited by 447 publications

(477 citation statements)

References 49 publications

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“…The proof that strong doubling implies (3.6) for Euclidean QHBC domains (contained in Lemma 4.5 and part of Theorem 3.10) extends without difficulty 1 to prove that strong doubling implies (6.2) when Ω is a metric ball (or more generally a metric John domain; see [7] and [16] for more details on such domains) in an HL-space. It is also readily verified that the global s-balance condition is satisfied if Ω is a metric ball (or metric John domain), dµ = δ s dλ, and dν = δ t dλ for some t ≥ 0.…”

Section: Sketch Of Proofmentioning

confidence: 99%

“…A large class of examples consists of the Carnot-Carathéodry spaces of Garofalo and Nhieu [16]. As usual with such spaces, the metric is defined in terms of sub-unit curves whose derivatives are almost everywhere linear combinations of certain Lipschitz continuous vector fields.…”

Section: Sketch Of Proofmentioning

confidence: 99%

“…The authors introduce hypotheses, labelled (H.1)-(H.3), which roughly say that the metric space is doubling at small scales, satisfies a "weak (1,1)-Poincaré inequality", and is a complete length space. Under the assumptions (H.1)-(H.3), inequalities (L ∞ ) and (Tr) are also true (see Theorems 1.9 and 1.11 of [16]); furthermore it is assumed that N = log 2 C d (so N equals the ambient dimension in the Euclidean context with Lebesgue measure).…”

Section: Sketch Of Proofmentioning

confidence: 99%

“…Several examples of Carnot-Carathéodory spaces satisfying hypotheses (H.1)-(H.3) are given in [16]. These examples include stratified nilpotent Lie groups and, more generally, connected Lie groups with polynomial volume growth, as well as spaces on which the vector fields are of Grushin or Hörmander type, and complete n-dimensional Riemannian manifolds with non-negative Ricci tensor.…”

Section: Sketch Of Proofmentioning

confidence: 99%

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Weighted Trudinger-type inequalities

Buckley¹,

O'Shea²

1999

Indiana Univ. Math. J.

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Abstract. We establish sharp inequalities of Trudinger-type (with non-standard Young functions) on general domains Ω with respect to measures ν, µ. Our results are new even when Ω is a Euclidean ball, and ν, µ are defined in terms of powers of distance to the boundary. In the Euclidean case, sharpness results are proved for the Young functions involved and the class of domains considered. New characterizations of QHBC domains are given.

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Section: Sketch Of Proofmentioning

confidence: 99%

Section: Sketch Of Proofmentioning

confidence: 99%

Section: Sketch Of Proofmentioning

confidence: 99%

Section: Sketch Of Proofmentioning

confidence: 99%

See 2 more Smart Citations

Weighted Trudinger-type inequalities

Buckley¹,

O'Shea²

1999

Indiana Univ. Math. J.

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“…We note that by Chow's accessibility theorem it follows, see [Ch], that the metric space (R n , d) is locally compact and that there exists R 0 > 0 such that the closure of any ball B cc ⊆ B cc (0, R 0 ) is compact. We stress that, in general, metric balls of large radii fail to be compact, see [GN1]. In view of these observations, we will in the following always assume that R 0 is such that…”

Section: Introductionmentioning

confidence: 99%

Wolff-potential estimates and doubling of subelliptic -harmonic measures

Avelin

Nyström

2013

Nonlinear Analysis: Theory, Methods & Applications

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Abstract. Let X = {X 1 , ..., Xm} be a system of C ∞ vector fields in R n satisfying Hörmander's finite rank condition and let Ω be a non-tangentially accessible domain with respect to the Carnot-Carathéodory distance d induced by X. We prove the doubling property of certain boundary measures associated to non-negative solutions, which vanish on a portion of ∂Ω, to the equationGiven p, 1 < p < ∞, fixed, we impose conditions on the function A = (A 1 , ..., Am) : R n × R m → R m , which imply that the equation is a quasi-linear partial differential equations of p-Laplace type structured on vector fields satisfying the classical Hörmander condition. In the case p = 2 and for linear equations, our result coincides with the doubling property of associated elliptic measures. To prove our result we establish, and this is of independent interest, a Wolff potential estimate for subelliptic equations of p-Laplace type.2000 Mathematics Subject Classification.

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Isoperimetric inequality from the poisson equation via curvature

Jiang

Koskela

2012

Comm Pure Appl Math

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In this paper, we establish an isoperimetric inequality in a metric measure space via the Poisson equation. Let (X,d,μ) be a complete, pathwise connected metric space with locally Ahlfors Q‐regular measure, where Q > 1, that supports a local L2‐Poincaré inequality. We show that, for the Poisson equation Δu = g, if the local L∞‐norm of the gradient Du can be bounded by the Lorentz norm LQ,1 of g, then we obtain an isoperimetric inequality and a Sobolev inequality in (X,d,μ) with optimal exponents. By assuming a suitable curvature lower bound, we establish such optimal bounds on $\||Du|\|_{L^\infty_{\rm loc}}$. © 2011 Wiley Periodicals, Inc.

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Isoperimetric and Sobolev inequalities for Carnot-Carath�odory spaces and the existence of minimal surfaces

Cited by 447 publications

References 49 publications

Weighted Trudinger-type inequalities

Weighted Trudinger-type inequalities

Wolff-potential estimates and doubling of subelliptic -harmonic measures

Isoperimetric inequality from the poisson equation via curvature

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