Geometric measure theory and differential inclusions

TL;DR

This paper investigates the regularity of stationary points of elliptic integrands in geometric measure theory, suggesting that classical regularity results like Allard's theorem may extend to broader classes of integrands.

Contribution

It shows that certain differential inclusions do not contain laminates, providing evidence that Allard's regularity theorem could hold for general elliptic integrands.

Findings

Differential inclusions considered do not contain laminates used in nonregular solutions.

Indication that Allard's theorem may extend to broader classes of elliptic integrands.

Open questions posed about regularity of stationary points for elliptic integrands.

Abstract

In this paper we consider Lipschitz graphs of functions which are stationary points of strictly polyconvex energies. Such graphs can be thought as integral currents, resp. varifolds, which are stationary for some elliptic integrands. The regularity theory for the latter is a widely open problem, in particular no counterpart of the classical Allard's theorem is known. We address the issue from the point of view of differential inclusions and we show that the relevant ones do not contain the class of laminates which are used in [22] and [25] to construct nonregular solutions. Our result is thus an indication that an Allard's type result might be valid for general elliptic integrands. We conclude the paper by listing a series of open questions concerning the regularity of stationary points for elliptic integrands.