An anisotropic Poincaré inequality in $GSB{V}^p$ and the limit of strongly anisotropic Mumford-Shah functionals

We show that functions in $GSB{V}^p$ in three-dimensional space with small variation in $2$ of $3$ directions are close to a function of one variable outside an exceptional set. Bounds on the volume and the perimeter in these two directions of the exceptional sets are provided. As a key tool we prove an approximation result for such functions by functions in $W^{1,p}$. For this we present a two-dimensional countable ball construction that allows to carefully remove the jumps of the function. As a direct application, we show $\mathrm{\Gamma }$-convergence of an anisotropic three-dimensional Mumford-Shah model to a one-dimensional model.