论文标题
步骤2的Carnot组中汉堡类型方程的分布解决方案
Distributional solutions of Burgers' type equations for intrinsic graphs in Carnot groups of step 2
论文作者
论文摘要
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 graph of a continuous function $φ\colon U\subseteq \mathbb W\to \mathbb L$ is $ c^1 _ {\ mathrm {h}} $ - 正常的是，当$φ$从分布意义上满足时，汉堡的类型系统$ d^φφ=ω$，带有连续的$ω$。我们强调的是，这种对等性并非在最简单的第三个Carnot小组中，即恩格尔集团。作为证明的工具，我们表明，连续的分配解决方案$φ$ $ d^φφ=ω$，$ω$连续，实际上是$ d^φφ=ω$的广泛解决方案。作为独立利息的副产品，我们获得了$ d^φφ=ω$的所有连续分配解决方案，$ω$连续，享受$ 1/2 $ - littlehölder的规律性沿垂直方向。
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 graph of a continuous function $φ\colon U\subseteq \mathbb W\to \mathbb L$ is $C^1_{\mathrm{H}}$-regular precisely when $φ$ satisfies, in the distributional sense, a Burgers' type system $D^φφ=ω$, with a continuous $ω$. We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group. As a tool for the proof we show that a continuous distributional solution $φ$ to a Burgers' type system $D^φφ=ω$, with $ω$ continuous, is actually a broad solution to $D^φφ=ω$. As a by-product of independent interest we obtain that all the continuous distributional solutions to $D^φφ=ω$, with $ω$ continuous, enjoy $1/2$-little Hölder regularity along vertical directions.
