  We continue our study in \cite{FL} on viscosity solutions to a one-phase free
boundary problem for the $p(x)$-Laplacian with non-zero right hand side. We
first prove that viscosity solutions are locally Lipschitz continuous, which is
the optimal regularity for the problem. Then we prove that Lipschitz free
boundaries of viscosity solutions are $C^{1,\alpha}$. We also present some
applications of our results.
  Moreover, we obtain new results for the operator under consideration that are
of independent interest, such as a Harnack inequality.
