#### Vol. 2, No. 3, 2020

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Structural analysis of an $L$-infinity variational problem and relations to distance functions

### Leon Bungert, Yury Korolev and Martin Burger

Vol. 2 (2020), No. 3, 703–738
##### Abstract

We analyse the functional $\mathsc{𝒥}\left(u\right)=\parallel \nabla u{\parallel }_{\infty }$ 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 $\mathsc{𝒥}$ and characterize the distance function as the unique nonnegative 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 the asymptotic profile of the gradient flow of $\mathsc{𝒥}$ and construct analytic solutions of fast marching type. In addition, we give a geometric characterization of the extreme points of the unit ball of $\mathsc{𝒥}$.

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 nonnegative eigenfunction on a graph.

##### Keywords
distance functions, nonlinear eigenfunctions, extreme points, gradient flows, weighted graphs
##### Mathematical Subject Classification 2010
Primary: 35P30, 47J10, 47J35, 49R05
Secondary: 05C12, 26A16