On ∞-Harmonic Functions on the Heisenberg Group

Bieske, Thomas

doi:10.1081/pde-120002872

Cited by 74 publications

(91 citation statements)

References 9 publications

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“…The idea has been made precise in [18]; see also [47], [64], [50], [39], [48], [19], [20], [13], [54] for further results. This approach requires much deeper prerequisites and, at first glance, seems to provide a solution to a different "absolute minimization" problem.…”

Section: And the Functionû Defined Bymentioning

confidence: 99%

“…Approximation of the Lipschitz extension problem by the sequence of functionals (8.4) as p → ∞ and taking a limit of the corresponding Euler-Lagrange equations were first proposed by Aronsson [5] and made completely rigorous, in case of the Euclidean norm, in [18]. Since then these ideas have been successfully employed in various settings and forms, e.g., in [13], [15], [19], [20], [39], [44], [47], [48], [50], [51], [54], [59], [60], and [64]. The facts about Sobolev spaces and functional analysis needed in order to prove the existence of p-harmonic functions by the direct method in calculus of variations can be found in many textbooks, see, e.g., [36] and [38].…”

Section: Regularitymentioning

confidence: 99%

“…Our presentation above follows [49]. In the special case of a sub-Riemannian setting, absolute minimizers and the associated generalized infinity Laplace equations have been considered in [19], [20], [21], [22] and [62]. The results obtained include uniqueness and equivalence of absolute minimizers and viscosity solutions of the infinity Laplace equation.…”

Section: Appendix: Absolute Minimizers In General Metric Spacesmentioning

confidence: 99%

“…Hence they did not treat the issues of this section in full generality. Related equations, which go outside of the framework of this manuscript, are found in [13], [22], [19], [28], [62] and [65]. Example 5.2 is new.…”

Section: The Infinity Laplace Operator In General Normsmentioning

confidence: 99%

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A tour of the theory of absolutely minimizing functions

Aronsson

Crandall

Juutinen

2004

Bull. Amer. Math. Soc.

296

382

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Abstract. These notes are intended to be a rather complete and self-contained exposition of the theory of absolutely minimizing Lipschitz extensions, presented in detail and in a form accessible to readers without any prior knowledge of the subject. In particular, we improve known results regarding existence via arguments that are simpler than those that can be found in the literature. We present a proof of the main known uniqueness result which is largely self-contained and does not rely on the theory of viscosity solutions. A unifying idea in our approach is the use of cone functions. This elementary geometric device renders the theory versatile and transparent. A number of tools and issues routinely encountered in the theory of elliptic partial differential equations are illustrated here in an especially clean manner, free from burdensome technicalities -indeed, usually free from partial differential equations themselves. These include a priori continuity estimates, the Harnack inequality, Perron's method for proving existence results, uniqueness and regularity questions, and some basic tools of viscosity solution theory. We believe that our presentation provides a unified summary of the existing theory as well as new results of interest to experts and researchers and, at the same time, a source which can be used for introducing students to some significant analytical tools.

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Section: And the Functionû Defined Bymentioning

confidence: 99%

Section: Regularitymentioning

confidence: 99%

Section: Appendix: Absolute Minimizers In General Metric Spacesmentioning

confidence: 99%

Section: The Infinity Laplace Operator In General Normsmentioning

confidence: 99%

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A tour of the theory of absolutely minimizing functions

Aronsson

Crandall

Juutinen

2004

Bull. Amer. Math. Soc.

296

382

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“…The only adjustment is to replace the Euclidean distance with the smooth Carnot norm N . See [3] for details in the case of the Heisenberg group.…”

Section: Carnot Jets and Viscosity Solutionsmentioning

confidence: 99%

A sub-Riemannian maximum principle and its application to the p-Laplacian in Carnot groups

Bieske¹

2012

Ann. Acad. Sci. Fenn. Math.

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Abstract. We prove a sub-Riemannian maximum principle for semicontinuous functions. We apply this principle to Carnot groups to provide a "sub-Riemannian" proof of the uniqueness of viscosity infinite harmonic functions. This is an alternate method of proof from the one found in [15]. We also establish the equivalence of weak solutions and viscosity solutions to the p-Laplace equation. This result extends the author's previous work in the Heisenberg group [3, 4]. Calculus on Carnot groupsWe begin by denoting an arbitrary Carnot group in R N by G and its corresponding Lie Algebra by g. Recall that g is nilpotent and stratified, resulting in the decompositionfor appropriate vector spaces that satisfy the Lie bracket relation [V 1 , V j ] = V 1+j . The Lie Algebra g is associated with the group G via the exponential map exp : g → G.Since this map is a diffeomorphism, we can choose a basis for g so that it is the identity map. Denote this basis byWe endow g with an inner product ·, · and related norm · so that this basis is orthonormal. Clearly, the Riemannian dimension of g (and so G) is N = n 1 +n 2 +n 3 . However, we will also consider the homogeneous dimension of G, denoted Q, which is given by

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A Visit with the ∞-Laplace Equation

Crandall

2008

Calculus of Variations and Nonlinear Partial Differential Equations

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On ∞-Harmonic Functions on the Heisenberg Group

Cited by 74 publications

References 9 publications

A tour of the theory of absolutely minimizing functions

A tour of the theory of absolutely minimizing functions

A sub-Riemannian maximum principle and its application to the p-Laplacian in Carnot groups

A Visit with the ∞-Laplace Equation

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