Regularity of flat free boundaries for two-phase p(x)-Laplacian problems with right hand side

We consider viscosity solutions to two-phase free boundary problems for the $p(x)$-Laplacian with non-zero right hand side. We prove that flat free boundaries are $C^{1,\gamma}$. No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the $p(x)$-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when $p(x)\equiv p$, i.e., for the $p$-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability.