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

Abstract

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 $S({\Omega }_1\times {\Omega }_2)$ 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 $S({\Omega }_1\times {\Omega }_2)$.