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
Abstract. We find the fundamental solution to the P -Laplace equation in Grushin-type spaces. The singularity occurs at the sub-Riemannian points, which naturally corresponds to finding the fundamental solution of a generalized Grushin operator in Euclidean space. We then use this solution to find an infinite harmonic function with specific boundary data and to compute the capacity of annuli centered at the singularity. A solution to the 2-Laplace equation in a wider class of spaces is presented.
Abstract. We derive the Euler-Lagrange equation (also known in this setting as the Aronsson-Euler equation) for absolute minimizers of thewhere Ω ⊂ G is an open subset of a Carnot group, ∇ 0 u denotes the horizontal gradient of u : Ω → R, and the Lipschitz class is defined in relation to the Carnot-Carathéodory metric. In particular, we show that absolute minimizers are infinite harmonic in the viscosity sense. As a corollary we obtain the uniqueness of absolute minimizers in a large class of groups. This result extends previous work of Jensen and of Crandall, Evans and Gariepy. We also derive the Aronsson-Euler equation for more "regular" absolutely minimizing Lipschitz extensions corresponding to those Carnot-Carathéodory metrics which are associated to "free" systems of vector fields.
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