BSDEs with monotone generator and two irregular reflecting barriers

We consider BSDEs with two reflecting irregular barriers. We give necessary and sufficient conditions for existence and uniqueness of ${\mathbb{L}}^p$ solutions for equations with generators monotone with respect to $y$ and Lipschitz continuous with respect to $z$, and with data in ${\mathbb{L}}^p$ spaces for $p\ge 1$. We also prove that the solutions can be approximated via penalization method.