Existence and uniqueness of a renormalized solution to fractional elliptic inclusions with L^1-data

Abstract

Let $d \in \Ni$ and $\Omega \subset \Ri^d$ be bounded with Lipschitz boundary. We prove the existence and uniqueness of a renormalized solution to the inclusion problem \[ \begin{cases} \beta(u) + Lu \ni f & \mbox{in } \Omega, \\ u = 0 & \mbox{in } \D\Omega, \end{cases} \] where $f \in L^1(\Omega)$, the operator $L$ generalizes the regional fractional $p$-Laplacian with $1 < p < \infty$ and $\beta$ is a maximal monotone operator on $\Ri$. A comparison between renormalized and weak solutions is also discussed.