On the constancy theorem for anisotropic energies through differential inclusions

TL;DR

This paper investigates stationary graphs for geometric functionals using differential inclusions, showing that non-negativity of the integrand prevents certain configurations, but dropping this condition allows constructing highly degenerate stationary points.

Contribution

It introduces a differential inclusion framework for stationary graphs with polyconvex integrands and characterizes the conditions under which certain configurations exist or are excluded.

Findings

Non-negativity of the integrand excludes T'_N configurations.

Dropping non-negativity allows constructing degenerate stationary points.

The main result generalizes previous work using differential inclusions.

Abstract

In this paper we study stationary graphs for functionals of geometric nature defined on currents or varifolds. The point of view we adopt is the one of differential inclusions, introduced in this context in the recent paper [De Lellis, De Philippis, Kirchheim, Tione, 2019]. In particular, given a polyconvex integrand $f$, we define a set of matrices $C_f$ that allows us to rewrite the stationarity condition for a graph with multiplicity as a differential inclusion. Then we prove that if $f$ is assumed to be non-negative, then in $C_f$ there is no $T'_N$ configuration, thus recovering the main result of [De Lellis, De Philippis, Kirchheim, Tione, 2019] as a corollary. Finally, we show that if the hypothesis of non-negativity is dropped, one can not only find $T'_N$ configurations in $C_f$, but it is also possible to construct via convex integration a very degenerate stationary point with multiplicity.