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We continue the analysis of a family of energies penalizing oscillations inoblique 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$\nabla_1\nabla_2u$ of functions with finite energy. The energies depend on aparameter $\theta\in(0,1]$ and the set of functions with finite energy growswith $\theta$. For $\theta<1$ we prove that the defect measure is$(n_1-1,n_2-1)$-tensor rectifiable in $\Omega_1\times\Omega_2$. We first getthe result for $n_1=n_2=1$ and deduce the general case through slicing usingWhite's rectifiability criterion. When $\theta=1$ the situation is less clearas measures of arbitrary dimensions from zero to $n_1+n_2-1$ are possible. Weshow however, in the case $n_1=n_2=1$ and for Lipschitz continuous functions,that the defect measures are $1\,$-rectifiable. This case bears stronganalogies with the study of entropic solutions of the eikonal equation.

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