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

We analyze a family of energies penalizing oscillations in oblique directions: they apply to functions u(x₁,x₂) with xₗ∈n_l and vanish when u(x) is of the form u₁(x₁) or u₂(x₂) . We mainly study the rectifiability properties of the defect measure ∇₁∇₂u of functions with finite energy. The energies depend on a parameter θ∈(0,1] . For θ<1 , we prove that the defect measure is (n₁-1,n₂-1) -tensor rectifiable. When instead θ=1 , we show, in the case n₁=n₂=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.