Regularity of the free boundary in Bernoulli-type problems

I study free boundary problems with Bernoulli-type free boundary conditions. These types of problems naturally appear in the study of the propagation of flames in the high activation energy limit. In the first part, a divergence-form degenerate elliptic operator is studied and the regularity of the free boundary is established. In the second part, a more natural development for regularity of the free boundary in the Bernoulli-type problem associated with the heat equation is proven. Finally, the uniqueness of solutions to general parabolic free boundary problems with Bernoulli-type free boundary conditions is shown