Size of characteristic sets and functions with prescribed gradient

Balogh, Zoltán M.

doi:10.1515/crll.2003.094

Cited by 87 publications

(132 citation statements)

References 26 publications

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“…In this paper, we generalize this result to the situation of general F (see Theorem D below and its proof in Section 6). For u ∈ C 1,1 and F = − X * in R 2n , Balogh showed (Theorem 3.1 (1) in [1]) that dim E S(u) < 2n− δ where dim E denotes the Hausdorff dimension with respect to the Euclidean metric and δ depends on the Lipschitz constant of ∇u. He also proved the existence of u ∈ ∩ 0<α<1 C 1,α such that S(u) has positive Lebesgue measure for any F ∈ C 1 (Ω) where Ω ⊂ R m is a given bounded domain (Theorem 4.1 (2) in [1]).…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Existence and uniqueness for P-area minimizers in the Heisenberg group

2006

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Abstract. In [3], we studied p-mean curvature and the associated p-minimal surfaces in the Heisenberg group from the viewpoint of PDE and differential geometry. In this paper, we look into the problem through the variational formulation. We study a generalized p-area and associated (p-) minimizers in general dimensions.We prove the existence and investigate the uniqueness of minimizers. Since this is reduced to solving a degenerate elliptic equation, we need to consider the effect of the singular set and this requires a careful study. We define the notion of weak solution and prove that in a certain Sobolev space, a weak solution is a minimizer and vice versa. We also give many interesting examples in dimension 2. An intriguing point is that, in dimension 2, a C 2 -smooth solution from the PDE viewpoint may not be a minimizer. However, this statement is true for higher dimensions due to the relative smallness of the size of the singular set.

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Existence and uniqueness for P-area minimizers in the Heisenberg group

2006

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“…By Sard theorem and the classical Whitney approximation theorem we can assume that a.e. level set is a submanifold of class C 1 , then we apply representation formula (2). The core of the proof stands in the key relation…”

Section: Introductionmentioning

confidence: 99%

Blow-up of regular submanifolds in Heisenberg groups and applications

Magnani

2006

Open Mathematics

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We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence of currents. Another application is the proof of an intrinsic coarea formula for vector-valued mappings on the Heisenberg group.

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“…In Corollary 1.8, it is important that the weak contact condition hold almost everywhere in the open set Ω, as the following example from [3] shows: Example 1.9. There is a continuously differentiable mapping f : R 2 → R 3 and a set E ⊆ R 2 such that im df x ⊆ ker α(f (x)) for all x ∈ E and H 2 H 1 (f (E)) > 0.…”

Section: Introductionmentioning

confidence: 99%

Weak contact equations for mappings into Heisenberg groups

Balogh¹,

Hajłasz²,

Wildrick³

2014

Indiana Univ. Math. J.

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Abstract. Let k > n be positive integers. We consider mappings from a subset of R k to the Heisenberg group H n with a variety of metric properties, each of which imply that the mapping in question satisfies some weak form of the contact equation arising from the sub-Riemannian structure of the Heisenberg group. We illustrate a new geometric technique that shows directly how the weak contact equation greatly restricts the behavior of the mappings. In particular, we provide a new and elementary proof of the fact that the Heisenberg group H n is purely k-unrectifiable. We also prove that for an open set Ω ⊂ R k , the rank of the weak derivative of a weakly contact mapping in the Sobolev space W 1,1 loc (Ω; R 2n+1 ) is bounded by n almost everywhere, answering a question of Magnani. Finally we prove that if f : Ω → H n is α-Hölder continuous, α > 1/2, and locally Lipschitz when considered as a mapping into R 2n+1 , then f cannot be injective. This result is related to a conjecture of Gromov.

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Size of characteristic sets and functions with prescribed gradient

Cited by 87 publications

References 26 publications

Existence and uniqueness for P-area minimizers in the Heisenberg group

Existence and uniqueness for P-area minimizers in the Heisenberg group

Blow-up of regular submanifolds in Heisenberg groups and applications

Weak contact equations for mappings into Heisenberg groups

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