Non-convex functionals penalizing simultaneous oscillations along two independent directions: Structure of the defect measure
Michael D. Goldman, Benoît Merlet · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2026
We analyze a family of energies penalizing oscillations in oblique directions: they apply to functions u(x_{1},x_{2}) with x_{l}\in\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 abla_{1} abla_{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.