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

We study a family of non-convex functionals {є} on the space of measurable functions u : Ω₁ × _Ω2 С R^n₁ × R^n₂ → R. These functionals vanish on the non-convex subset S(Ω₁ × Ω₂) formed by functions of the form u(x₁, x₂) = u₁(x₁) or u(x₁, x₂) = u₂(x₂). We investigate under which conditions the converse implication “є(u) = 0 ⇒ ϵ 2 S(Ω₁ × Ω₂)” 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) є(u) controls in a strong sense the distance of u to S(Ω_{1 ×} Ω₂).