Distributional Solutions of Burgers’ type Equations for Intrinsic Graphs in Carnot Groups of Step 2

Abstract

We prove that in arbitrary Carnot groups $\mathbb {G}$ G of step 2, with a splitting $\mathbb {G}=\mathbb {W}\cdot \mathbb {L}$ G = W ⋅ L with $\mathbb {L}$ L one-dimensional, the intrinsic graph of a continuous function $\varphi \colon U\subseteq \mathbb {W}\to \mathbb {L}$ φ : U ⊆ W → L is $C^{1}_{\mathrm {H}}$ C H 1 -regular precisely when φ satisfies, in the distributional sense, a Burgers’ type system Dφφ = ω, with a continuous ω. We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group. We notice that our results generalize previous works by Ambrosio-Serra Cassano-Vittone and Bigolin-Serra Cassano in the setting of Heisenberg groups. As a tool for the proof we show that a continuous distributional solution φ to a Burgers’ type system Dφφ = ω, with ω continuous, is actually a broad solution to Dφφ = ω. As a by-product of independent interest we obtain that all the continuous distributional solutions to Dφφ = ω, with ω continuous, enjoy 1/2-little Hölder regularity along vertical directions.

Area formula for regular submanifolds of low codimension in Heisenberg groups

“…Connecting the Jacobian with the differential, and hence allowing for an effective computation of the Hausdorff (or spherical) measure of a set, has been completely achieved for intrinsic regular hypersurfaces in stratified groups. This result stems from the contribution of many authors; see [4] for the last version of this one-codimensional area formula, along with the full list of references. For one-codimensional intrinsic Lipschitz graphs, area formulas for the spherical measure are obtained in [11] for stratified groups of step two; see also the references therein.…”

“…Although the parametrizing mapping of the ℍ-regular surface is not Lipschitz continuous in the Euclidean sense, in [6] Arena and Serapioni proved that it is uniformly intrinsically differentiable (Definition 2.15). Indeed, uniform intrinsic differentiability for maps acting between suitable factorizing homogeneous subgroups has been largely studied, also in a broader framework and from the viewpoint of nonlinear first order systems of PDEs [3,4,9,12,23].…”