Abstract
We characterize $p-$harmonic functions in the Heisenberg group in terms of an asymptotic mean value property, where $1 < p <\infty$, following the schemedescribed in [16] for the Euclidean case. The new tool that allows us to consider the subelliptic case is a geometric lemma, Lemma 3.2 below, that relates the directions of the points of maxima and minima of a function on a small subelliptic ball with the unit horizontal gradient of that function.
Highlights
In this paper we study p−harmonic functions in the Heisenberg group in terms of an asymptotic mean value property
More precisely in [MPR], the authors show that if u is a continuous function in a domain Ω ⊂ Rn and p ∈ (1, ∞], the asymptotic expansion u(x) = p − 2 max u + min u + 2 + n u(y) dy + o(ǫ2), 2(p + n) Bǫ(x)
We want to extend this characterization to functions defined on the Heisenberg group Hn
Summary
In this paper we study p−harmonic functions in the Heisenberg group in terms of an asymptotic mean value property. P-Laplacian, Heisenberg group, Mean value formulas, Viscosity solutions. As shown in [JLM] for the Euclidean case and in [Bi] for the subelliptic case, it suffices to consider smooth functions whose horizontal gradient does not vanish In addition in those papers it is shown that the notions of viscosity and weak solutions agree for homogeneous equation −∆p,Hnu = 0. As ǫ → 0 in viscosity sense, if (i) for every continuous function φ defined in a neighborhood of a point P such that u − φ has a strict minimum at P with u(P ) = φ(P ) we have. We certainly conjecture that the statement of Theorem 1.1 holds in this case
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