Lipschitz continuity, Aleksandrov theorem and characterizations for H-convex functions

Magnani, Valentino

doi:10.1007/s00208-005-0717-4

Cited by 42 publications

(56 citation statements)

References 28 publications

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“…To quote a few: h 0 can be defined in terms of the first variation of the area functional [16,15,19,26,37,13], as horizontal divergence of the horizontal unit normal or as limit of the mean curvatures h ǫ of suitable Riemannian approximating metrics σ ǫ [13]. If the surface is not regular, the notion of curvature can be expressed in the viscosity sense (we refer to [3], [4], [42], [41], [29], [2], [31], [6] for viscosity solutions of PDE in the sub-Riemannian setting).…”

Section: 2mentioning

confidence: 99%

Regularity of mean curvature flow of graphs on Lie groups free up to step 2

2015

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Abstract. We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics σ ǫ collapsing to a subRiemannian metric σ 0 as ǫ → 0. We establish C k,α estimates for this flow, that are uniform as ǫ → 0 and as a consequence prove long time existence for the subRiemannian mean curvature flow of the graph. Our proof extend to the setting of every step two Carnot group (not necessarily free) and can be adapted following our previous work in [10] to the total variation flow. IntroductionThe mean curvature flow is the motion of a surface where each points is moving in the direction of the normal with speed equal to the mean curvature In the case where the evolution of graphs S t = {(x, u(x, t))} ⊂ R n × R is considered, then, provided enough regularity is assumed, the function u satisfies the equationGiven appropriate boundary/initial conditions, global in time solutions asymptotically converge to minimal graphs.In this paper we study long time existence of graph solutions of the mean curvature flow in a special class of degenerate Riemannian ambient spaces: The so-called sub-Riemannian Hörmander type setting [22], [40]. In particular we will focus on a class of Lie groups endowed with a metric structure (G, σ 0 ) that arises as limit of collapsing left-invariant tame Riemannian structures (G, σ ǫ ).Our approach to the existence of global (in time) smooth solutions is based on a Riemannian approximation scheme. We study graph solutions of the mean curvature flow in the Riemannian spaces (G, σ ǫ ) where G is a group and σ ǫ is a family of Riemannian metrics that 'collapse' as ǫ → 0 to a sub-Riemannian metric σ 0 in G.

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Section: 2mentioning

confidence: 99%

Regularity of mean curvature flow of graphs on Lie groups free up to step 2

2015

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“…On the Heisenberg groups, these notions coincide (Balogh and Rickly [1]), whereas, generally, the v-convex functions are the u.s.c. h-convex functions (see Magnani [23], Wang [32]). In this paper we are concerned with the notion of v-convexity.…”

Section: Introductionmentioning

confidence: 99%

A new characterization of convexity in free Carnot groups

Bonfiglioli

Lanconelli

2012

Proc. Amer. Math. Soc.

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Abstract. A characterization of convex functions in R N states that an upper semicontinuous function u is convex if and only if u(Ax) is subharmonic (with respect to the usual Laplace operator) for every symmetric positive definite matrix A. The aim of this paper is to prove that an analogue of this result holds for free Carnot groups G when considering convexity in the viscosity sense. In the subelliptic context of Carnot groups, the linear maps x → Ax of the Euclidean case must be replaced by suitable group isomorphisms x → T A (x), whose differential preserves the first layer of the stratification of Lie(G).

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“…This question has not yet been addressed in the sub-Riemannian setting, where however a number of authors have studied viscosity solutions for non degenerate PDE: [8], [9], [65], [66], [47], [2], and [49].…”

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confidence: 99%