论文标题
$ l $ infinity差异问题的结构分析以及与距离功能的关系
Structural analysis of an $L$-infinity variational problem and relations to distance functions
论文作者
论文摘要
在这项工作中，我们分析了功能性$ {\ cal j}（u）= \ | \ nabla u \ | _ \ | _ \ infty $在Lipschitz上定义的具有均匀的dirichlet边界条件的功能。我们的分析直接在功能上执行，而无需使用平滑的$ p $ norms进行近似。我们证明其接地状态与距离函数到域边界的倍数一致。此外，我们计算$ {\ cal j} $的$ l^2 $ -subDifferential，并将距离函数描述为子差异操作员的唯一非负eigenFunction。我们还研究了一般本征函数的特性，尤其是它们的淋巴结集。此外，我们证明可以将距离函数计算为$ {\ cal j} $的梯度流的渐近概况，并构造快速行进类型的分析解决方案。此外，我们给出$ {\ cal j} $的单位球的极端点的几何表征。 最后，我们将其中许多结果传输到有限加权图上定义的功能的离散版本。在这里，我们分析了距离函数在图及其梯度上的属性。连续体和离散设置之间的主要区别在于，距离函数不是图上唯一的非负征函数。
In this work we analyse the functional ${\cal J}(u)=\|\nabla u\|_\infty$ defined on Lipschitz functions with homogeneous Dirichlet boundary conditions. Our analysis is performed directly on the functional without the need to approximate with smooth $p$-norms. We prove that its ground states coincide with multiples of the distance function to the boundary of the domain. Furthermore, we compute the $L^2$-subdifferential of ${\cal J}$ and characterize the distance function as unique non-negative eigenfunction of the subdifferential operator. We also study properties of general eigenfunctions, in particular their nodal sets. Furthermore, we prove that the distance function can be computed as asymptotic profile of the gradient flow of ${\cal J}$ and construct analytic solutions of fast marching type. In addition, we give a geometric characterization of the extreme points of the unit ball of ${\cal J}$. Finally, we transfer many of these results to a discrete version of the functional defined on a finite weighted graph. Here, we analyze properties of distance functions on graphs and their gradients. The main difference between the continuum and discrete setting is that the distance function is not the unique non-negative eigenfunction on a graph.
