  The $p$-Laplacian is a nonlinear partial differential equation, parametrized
by $p \in [1,\infty]$. We provide new numerical algorithms, based on the
barrier method, for solving the $p$-Laplacian numerically in $O(\sqrt{n}\log
n)$ Newton iterations for all $p \in [1,\infty]$, where $n$ is the number of
grid points. We confirm our estimates with numerical experiments.
