论文标题
单调在度量表面上功能的coarea不平等
Coarea Inequality for Monotone Functions on Metric Surfaces
论文作者
论文摘要
我们研究了公制表面的Coarea不平等现象 - 拓扑表面的度量空间，没有边界，并且具有局部有限的Hausdorff 2-Measure $ \ MATHCAL {H}^2 $。对于单调sobolev函数$ u \ u \ colon x \ to \ mathbb {r} $，我们证明了不等式 \ begin {equation*} \ int_ {\ mathbb {r}}}^{*} \ int_ {u^{ - 1}（t）} g \，D \ Mathcal {H}^{1} \，dt \ leq κ \ int_ {x} Gρ \，d \ Mathcal {h}^{2} \ quad \ text {对于每个borel $ g \ colon x \ rightarrow \ left [0，\ infty \ right] $，} \ end {equation*}，其中$ρ$是$ u $的任何可集成的上梯度。如果$ρ$是本地$ l^2 $ - 积分，我们将获得尖锐的常数$κ= 4/π$。单调性条件无法删除，因为我们给出了公制表面$ x $的示例和lipschitz函数$ u \ u \ colon x \ to \ mathbb {r} $，因此上述coarea不平等是失败的。
We study coarea inequalities for metric surfaces -- metric spaces that are topological surfaces, without boundary, and which have locally finite Hausdorff 2-measure $\mathcal{H}^2$. For monotone Sobolev functions $u\colon X \to \mathbb{R} $, we prove the inequality \begin{equation*} \int_{ \mathbb{R} }^{*} \int_{ u^{-1}(t) } g \,d\mathcal{H}^{1} \,dt \leq κ \int_{ X } g ρ \,d\mathcal{H}^{2} \quad\text{for every Borel $g \colon X \rightarrow \left[0,\infty\right]$,} \end{equation*} where $ρ$ is any integrable upper gradient of $u$. If $ρ$ is locally $L^2$-integrable, we obtain the sharp constant $κ=4/π$. The monotonicity condition cannot be removed as we give an example of a metric surface $X$ and a Lipschitz function $u \colon X \to \mathbb{R}$ for which the coarea inequality above fails.
