On the constancy theorem for anisotropic energies through 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 papers (De Lellis et al. in Geometric measure theory and differential inclusions, 2019. arXiv:1910.00335; Tione in Minimal graphs and differential inclusions. Commun Part Differ Equ 7:1–33, 2021). In particular, given a polyconvex integrand f, we define a set of matrices $$C_f$$ 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 $$C_f$$ C f there is no $$T'_N$$ T N ′ configuration, thus recovering the main result of De Lellis et al. (Geometric measure theory and differential inclusions, 2019. arXiv:1910.00335) as a corollary. Finally, we show that if the hypothesis of non-negativity is dropped, one can not only find $$T'_N$$ T N ′ configurations in $$C_f$$ C f , but it is also possible to construct via convex integration a very degenerate stationary point with multiplicity.

Critical points of degenerate polyconvex energies

“…which is equivalent to div(Du T Df (Du) − f (Du) id) = 0 in the weak sense. The situation for stationary points to certain polyconvex functionals seems to be more rigid than the one for critical points, see [5,6,12,27,28]. This is the case also in the problem considered in this paper.…”

The anisotropic min‐max theory: Existence of anisotropic minimal and CMC surfaces

“…We explicitly observe that () does not depend on $$c$$. For a more detailed study about anisotropic CMC surfaces in the Euclidean space, we refer the author to [13, 15, 19–21, 23–25, 29]. Remark We remark that we can absorb the metric of $$M$$ in the elliptic integrand $$G$$.…”

Non-classical solutions of the $p$-Laplace equation

“…Similar techniques were developed to find counterexamples in which oscillation phenomena are the issue. Namely, while the staircase laminate construction deals, roughly speaking, with finding maps which are in some W 1,p space but no better, similar methods can be used, for instance, to find maps which Lipschitz and not C 1 on any open set, see for instance [6,18,30,32,33]. Instead of outlining our proof here we defer the discussion to Section 3, where the structure of the part of the paper devoted to the construction of the counterexample will be explained in detail.…”