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

Abstract

AbstractWe prove that in arbitrary Carnot groups$\mathbb {G}$Gof step 2, with a splitting$\mathbb {G}=\mathbb {W}\cdot \mathbb {L}$G=W⋅Lwith$\mathbb {L}$Lone-dimensional, the intrinsic graph of a continuous function$\varphi \colon U\subseteq \mathbb {W}\to \mathbb {L}$φ:U⊆W→Lis$C^{1}_{\mathrm {H}}$CH1-regular precisely whenφsatisfies, in the distributional sense, a Burgers’ type systemDφφ=ω, with a continuousω. 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 solutionφto a Burgers’ type systemDφφ=ω, withωcontinuous, is actually a broad solution toDφφ=ω. As a by-product of independent interest we obtain that all the continuous distributional solutions toDφφ=ω, withωcontinuous, enjoy 1/2-little Hölder regularity along vertical directions.