*Accepted Paper*

**Inserted:** 20 may 2019

**Last Updated:** 26 may 2020

**Journal:** Annali della SNS

**Year:** 2020

**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 $S(\Omega_1\times\Omega_2)$.

**Download:**