Regularity of flat free boundaries for two-phase \(p(x)\)-Laplacian problems with right hand side (Q6545033)

The paper studies viscosity solutions to two-phase free boundary problems involving the \(p(x)\)-Laplacian with a non-zero right-hand side. The authors establish that flat free boundaries are \(C^{1,\gamma}\), without assuming Lipschitz continuity of the solutions. These findings represent the first regularity results in the literature for such problems, even when \(p(x) \equiv\) constant, applicable to both singular and degenerate operators. By applying to viscosity solutions, the results offer wide applicability and mark a significant advancement in the understanding of free boundary problems associated with the \(p(x)\)-Laplacian.