We investigate the regularity of the free boundary for a general class of two-phase free boundary problems with nonzero right-hand side. We prove that Lipschitz or flat free boundaries are . In particular, viscosity solutions are indeed classical.
"Two-phase problems with distributed sources: regularity of the free boundary." Anal. PDE 7 (2) 267 - 310, 2014. https://doi.org/10.2140/apde.2014.7.267