Abstract
We prove that in arbitrary Carnot groups \(\mathbb {G}\) of step 2, with a splitting \(\mathbb {G}=\mathbb {W}\cdot \mathbb {L}\) with \(\mathbb {L}\) one-dimensional, the intrinsic graph of a continuous function \(\varphi \colon U\subseteq \mathbb {W}\to \mathbb {L}\) is \(C^{1}_{\mathrm {H}}\)-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.
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D.D.D., S.D. were partially supported by the Academy of Finland (grant 288501 ‘Geometry of subRiemannian groups’ and by grant 322898 ‘Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory’). G.A., D.D.D., S.D. were partially supported by the European Research Council (ERC Starting Grant 713998 GeoMeG ‘Geometry of Metric Groups’). S. Don was also supported by the Swiss National Science Foundation, Grant Nr. 200020 191978. The authors wish to express their gratitude to the anonymous referee for useful feedback on the manuscript.
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Antonelli, G., Di Donato, D. & Don, S. Distributional Solutions of Burgers’ type Equations for Intrinsic Graphs in Carnot Groups of Step 2. Potential Anal 59, 1481–1506 (2023). https://doi.org/10.1007/s11118-022-09992-x
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DOI: https://doi.org/10.1007/s11118-022-09992-x
Keywords
- Carnot groups
- Step-2 Carnot groups
- Intrinsically C 1-surfaces
- Broad solutions
- Burgers’ equation
- Distributional solutions to non-linear first order PDEs