compactness Theorem \[teo:sigma\_equicoercive\].

The fact that we are working in dimension $2$ is fundamental for the proof of Theorem \[teo:sigma\_liminf\] as it allows to locally rewrite the vector field $\s_{\varepsilon}$ as the rotated gradient of a function.

**Structure of the paper:** In Section \[sec:Not\] we introduce some notation and several tools and notions on $SBV$ functions and vector field measures. In Section \[sec:LocRes\] we study a first family of energies obtained by substituting ${\nabla}u$ for $\s$ in the definition of ${\mathcal{F}}\e$. In Section \[sec:Comp\] we prove the equicoercivity result, Theorem \[teo:sigma\_equicoercive\] and we establish the lower bound stated in Theorem \[teo:sigma\_liminf\]. In Section \[sec:UppBound\] we prove the upper bound of Theorem \[teo:sigma\_limsup\]. Finally, in the last section, we present and discuss various numerical simulations.

Notation and Preliminary Results {#sec:Not}
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In the following ${\Omega}\subset\subset\hat{\Omega}\subset{\mathbf{R}}^d$ are bounded open convex sets. Given $X\subset{\mathbf{R}}^d$ (in practice $X={\Omega}$ or $X=\hat{\Omega})$, we denote by $\A(X)$ the class of all open subsets of $X$ and by $\A_S(X)$ the subclass of all simply connected open sets $O\subset X$ such that $\overline{O}\cap S=\varnothing$. We denote by $(e_1,\dots,e_d)$ the canonical orthonormal basis of ${\mathbf{R}}^d$, by $|\cdot|$ the euclidean norm and by $\langle\cdot,\cdot\rangle$) the euclidean scalar product in ${\mathbf{R}}^d$. The 