Non-convex functionals penalizing simultaneous oscillations along independent directions: rigidity estimates

We study a family of non-convex functionals {E} on the space of measurable functions u : Omega(1) x Omega(2) subset of R-n1 x R-n2 -> R. These functionals vanish on the non-convex subset S(Omega(1) x Omega(2)) formed by functions of the form u(x(1), x(2)) = u(1)(x(1)) or u(x(1), x(2)) = u(2)(x(2)). We investigate under which conditions the converse implication "E(u) = 0 double right arrow u is an element of S(Omega(1) x Omega(2))" holds. In particular, we show that the answer depends strongly on the smoothness of u. We also obtain quantitative versions of this implication by proving that (at least for some parameters) E(u) controls in a strong sense the distance of u to S(Omega(1) x Omega(2)).