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

## TL;DR

This paper investigates non-convex functionals that penalize simultaneous oscillations in functions of two variables, establishing conditions under which zero functional value implies the function is separable, with quantitative estimates depending on smoothness.

## Contribution

It provides new rigidity estimates for non-convex functionals, linking zero energy to function structure and smoothness, with quantitative control of the distance to separable functions.

## Key findings

- Zero functional value implies separability under certain smoothness conditions.
- Quantitative bounds relate the functional value to the distance from the set of separable functions.
- Smoothness of the function critically influences the 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 $S(\Omega_1\times\Omega_2)$.