Regularity results for a two-phase free boundary problem for an inhomogeneous equation with non-standard growth

We consider viscosity solutions to a two-phase free boundary problem for a nonlinear elliptic PDE with non-zero right hand side. We obtain regularity results for solutions and their free boundaries. The operator under consideration is a model case in the class of partial differential equations with non-standard growth. This type of operators have been used in the modelling of non-Newtonian fluids, such as electrorheological or thermorheological fluids, also in non-linear elasticity and image reconstruction. The fact that we are dealing with nonlinear degenerate/singular equations with non-zero right hand side leads to challenging difficulties that will be addressed in this talk. This is joint work with Fausto Ferrari (University of Bologna, Italy)