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 \(C^{1,\gamma }\) $C^{1,\gamma }$ . 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 \(p(x)\equiv p\) $p(x)\equiv p$ , i.e., for the p-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability.