Non-convex functionals penalizing simultaneous oscillations along two independent directions: Structure of the defect measure

Abstract

We analyze a family of energies penalizing oscillations in oblique directions: they apply to functions $u(x_1,x_2)$ with $x_l\in {\mathbb{R}}^{n_l}$ and vanish when $u(x)$ is of the form $u_1(x_1)$ or $u_2(x_2)$. We mainly study the rectifiability properties of the defect measure ${\nabla }_1{\nabla }_{2}u$ of functions with finite energy. The energies depend on a parameter $\theta \in (0,1]$. For $\theta <1$, we prove that the defect measure is $(n_1-1,n_2-1)$-tensor rectifiable. When instead $\theta =1$, we show, in the case $n_1=n_2=1$ and for Lipschitz continuous functions, that the defect measures are $1$-rectifiable. This case bears strong analogies with the study of entropic solutions of the eikonal equation and can be recast as a differential inclusion.