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

Abstract 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, C 1, γ. 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) p p (x) ≡ p, i. e. , for the p -Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability.