This thesis consists of an introduction and four research papers related to free boundary problems and systems of fully non-linear elliptic equations.
Paper A and Paper B prove optimal regularity of solutions to general elliptic and parabolic free boundary problems, where the operators are fully non-linear and convex. Furthermore, it is proven that the free boundary is continuously differentiable around so called "thick" points, and that the free boundary touches the fixed boundary tangentially in two dimensions.
Paper C analyzes singular points of solutions to perturbations of the unstable obstacle problem, in three dimensions. Blow-up limits are characterized and shown to be unique. The free boundary is proven to lie close to the zero-level set of the corresponding blow-up limit. Finally, the structure of the singular set is analyzed.
Paper D discusses an idea on how existence and uniqueness theorems concerning quasi-monotone fully non-linear elliptic systems can be extended to systems that are not quasi-monotone.