Lower bounds and Poincaré inequality for the Brezis-Seeger-Van Schaftingen-Yung functionals

We consider a family of non-local and non-convex functionals depending on three real parameters $(\gamma ,p,\lambda )$, which was introduced by H. Brezis, A. Seeger, J. Van Schaftingen and P. L. Yung. In the case $\gamma <-p$ we prove a lower bound for the Gamma-liminf of these functionals as $\lambda \to 0^{+}$ and a Poincaré-type inequality. We also show that such inequality cannot hold if $\gamma >0$, and we establish a modified version in this case. These results answer some open questions posed by these authors and by H. M. Nguyen.