Calculus of Variations and Geometric Measure Theory

A. De Rosa - Y. Lei - R. Young

Construction of fillings with prescribed Gaussian image and applications

created by derosa on 22 Jan 2024

[BibTeX]

preprint

Inserted: 22 jan 2024
Last Updated: 22 jan 2024

Year: 2024

ArXiv: 2401.10858 PDF

Abstract:

We construct $d$-dimensional polyhedral chains such that the distribution of tangent planes is close to a prescribed measure on the Grassmannian and the chains are either cycles (if the prescribed measure is centered) or their boundary is the boundary of a unit $d$-cube (if the barycenter of the prescribed measure, considered as a measure on $\bigwedge^d \mathbb{R}^n$, is a simple $d$-vector). Such fillings were first proved to exist by Burago and Ivanov (Geom. funct. anal., 2004); our work gives an explicit construction. Furthermore, in the case that the measure on the Grassmannian is supported on the set of positively oriented $d$-planes, we can construct fillings that are Lipschitz multigraphs. We apply this construction to prove that for anisotropic surface energies, ellipticity for Lipschitz multivalued functions is equivalent to polyconvexity and to show that strict polyconvexity is necessary for the atomic condition to hold.