Calculus of Variations and Geometric Measure Theory

J. Hirsch - R. Tione

On the constancy theorem for anisotropic energies through differential inclusions

created by tione on 28 Nov 2020
modified on 07 Oct 2021


Published Paper

Inserted: 28 nov 2020
Last Updated: 7 oct 2021

Journal: Calc. Var. Partial Differential Equations
Year: 2020

ArXiv: 2010.14846 PDF


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.