论文标题
$ \ mathbb h^1 $的常规和结晶规范的等级问题
The isoperimetric problem for regular and crystalline norms in $\mathbb H^1$
论文作者
论文摘要
我们研究了$ \ mathbb r^3 $的各向异性左转外周长测量的等等问题，并具有Heisenberg组的结构。外围与水平分布上的剩余不变标准$ ϕ $相关联。我们首先证明了定期套装的$ ϕ $ - 过时的表示公式，并假设$ ϕ $有一些规律性，并且在其双范围$ ϕ^*$上，我们根据$ \ mathrm c^2 $ -Smotherface的sub-finsler Geodesics推定了叶片属性，具有固定的$ -Culface。然后，我们证明了$ \ mathrm c^2 $ - 平滑表面的特征集，这些表面是当地极端的，对于等级问题，是由孤立的点和满足合适微分方程的水平曲线组成的。基于这样的特征，我们将$ \ mathrm c^2 $ -smooth $ ϕ $ -isoperimetric套装作为潘森气泡的子鳍类似物。我们还显示，在$ ϕ $上的适当规律性属性下，该子捕集者候选人的等级集确实是$ \ mathrm c^2 $ -smooth。通过近似过程，我们最终证明了在一般情况下（包括$ ϕ $是结晶的情况）的候选解决方案的条件最小属性。
We study the isoperimetric problem for anisotropic left-invariant perimeter measures on $\mathbb R^3$, endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm $ϕ$ on the horizontal distribution. We first prove a representation formula for the $ϕ$-perimeter of regular sets and, assuming some regularity on $ϕ$ and on its dual norm $ϕ^*$, we deduce a foliation property by sub-Finsler geodesics of $\mathrm C^2$-smooth surfaces with constant $ϕ$-curvature. We then prove that the characteristic set of $\mathrm C^2$-smooth surfaces that are locally extremal for the isoperimetric problem is made of isolated points and horizontal curves satisfying a suitable differential equation. Based on such a characterization, we characterize $\mathrm C^2$-smooth $ϕ$-isoperimetric sets as the sub-Finsler analogue of Pansu's bubbles. We also show, under suitable regularity properties on $ϕ$, that such sub-Finsler candidate isoperimetric sets are indeed $\mathrm C^2$-smooth. By an approximation procedure, we finally prove a conditional minimality property for the candidate solutions in the general case (including the case where $ϕ$ is crystalline).
