Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations

Liu, Qing; Zhou, Xiaodan

doi:10.2422/2036-2145.201907_001

Cited by 3 publications

(7 citation statements)

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“…It turns out that in general one can obtain the quasiconvex envelope by iterating the operator T . Such type of iteration is also used in [21] to construct the h-convex envelope of a given continuous function in the Heisenberg group. Theorem 2.12 (Iterative scheme with direct convexification).…”

Section: H-quasiconvex Envelopementioning

confidence: 99%

“…Various properties and generalizations of such convex functions are discussed in [2,18,30,23,8,3,24] etc. The corresponding convex envelope and its applications to convexity properties of sub-elliptic equations are studied in [21].…”

mentioning

confidence: 99%

“…It turns out that T [f ] itself may not be h-quasiconvex in Ω but iterated application of T yields a pointwise approximation of the h-quasiconvex envelope Q(f ); see Theorem 2.12 for details. This type of construction is also adopted in [21] for the h-convex envelope of a given function f .…”

mentioning

confidence: 99%

“…It is not clear to us whether one can replace dH by the left invariant gauge metric d H . This problem is related to the h-convexity preserving property for solutions of evolution equations in the Heisenberg group; see some partial results in [20,21].…”

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confidence: 99%

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Horizontally quasiconvex envelope in the Heisenberg group

Kijowski¹,

Liu²,

Zhou³

2022

Preprint

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This paper is concerned with a PDE-based approach to the horizontally quasiconvex (h-quasiconvex for short) envelope of a given continuous function in the Heisenberg group. We provide a characterization for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to a first-order nonlocal Hamilton-Jacobi equation. We also construct the corresponding envelope of a continuous function by iterating the nonlocal operator. One important step in our arguments is to prove the uniqueness and existence of viscosity solutions to the Dirichlet boundary problems for the nonlocal Hamilton-Jacobi equation. Applications of our approach to the h-convex hull of a given set in the Heisenberg group are discussed as well.

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Section: H-quasiconvex Envelopementioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Horizontally quasiconvex envelope in the Heisenberg group

Kijowski¹,

Liu²,

Zhou³

2022

Preprint

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“…Thus, a tightly related question is the study of the regularity of supersolutions satisfying convexity constraints and linear or nonlinear partial differential inequalities in nondivergence form. In this direction, a study of the horizontal convex envelope in the Heisenberg group, along with its application to the study of horizontal convexity properties of solutions to fully nonlinear equations was performed in [28]. This analysis has its roots in the earlier work by Alvarez, Lasry, and Lions [4].…”

Section: Introductionmentioning

confidence: 99%

Interior a priori estimates for supersolutions of fully nonlinear subelliptic equations under geometric conditions

Goffi

2024

Bulletin of London Math Soc

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In this note, we prove interior a priori first‐ and second‐order estimates for solutions of fully nonlinear degenerate elliptic inequalities structured over the vector fields of Carnot groups, under the main assumption that is semiconvex along the fields. These estimates for supersolutions are new even for linear subelliptic inequalities in nondivergence form, whereas in the nonlinear setting they do not require neither convexity nor concavity on the second derivatives. We complement the analysis exhibiting an explicit example showing that horizontal regularity of Calderón–Zygmund type for fully nonlinear subelliptic equations posed on the Heisenberg group cannot be in general expected in the range , being the homogeneous dimension of the group.

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Horizontally quasiconvex envelope in the Heisenberg group

2024

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This paper is concerned with a PDE approach to horizontally quasiconvex (h-quasiconvex) functions in the Heisenberg group based on a nonlinear second order elliptic operator. We discuss sufficient conditions and necessary conditions for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to the associated elliptic equation. Since the notion of h-quasiconvexity is equivalent to the horizontal convexity (h-convexity) of the function's sublevel sets, we further adopt these conditions to study the h-convexity preserving property for horizontal curvature flow in the Heisenberg group. Under the comparison principle, we show that the curvature flow starting from a star-shaped h-convex set preserves the h-convexity during the evolution.

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Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations

Cited by 3 publications

References 29 publications

Horizontally quasiconvex envelope in the Heisenberg group

Horizontally quasiconvex envelope in the Heisenberg group

Interior a priori estimates for supersolutions of fully nonlinear subelliptic equations under geometric conditions

Horizontally quasiconvex envelope in the Heisenberg group

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