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

We study a family of non-convex functionals $\{\mathcal{E}\}$ on the space of measurable functions$u: \Omega_1\times\Omega_2 \subset $\mathbb{R}$^{n_1}\times$\mathbb{R}$^{n_2} \to $\mathbb{R}$$. These functionals vanish on the non-convex subset $S({\mathrm{\Omega }}_1\times {\mathrm{\Omega }}_2)$ formed by functions of the form $u(x_1,x_2)=u_1(x_1)$ or $u(x_1,x_2)=u_2(x_2)$. We investigate under which conditions the converse implication “$$\mathcal{E}$(u) = 0 \Rightarrow u \in S(\Omega_1\times\Omega_2)$” 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) $\mathcal{E}(u)$ controls in a strong sense the distance of $u$ to $S({\mathrm{\Omega }}_1\times {\mathrm{\Omega }}_2)$.