The free boundary for semilinear problems with highly oscillating singular terms

We investigate general semilinear (obstacle-like) problems of the form $\Delta u = f(u)$, where $f(u)$ has a singularity/jump at $\{u=0\}$ giving rise to a free boundary. Unlike many works on such equations where $f$ is approximately homogeneous near $u = 0$, we work under assumptions allowing for highly oscillatory behavior. We establish the $C^\infty$ regularity of the free boundary $\partial \{u>0\}$ at flat points. Our approach is to first establish that flat free boundaries are Lipschitz, using a comparison argument with the Kelvin transform. For higher regularity, we study the highly degenerate PDE satisfied by ratios of derivatives of $u$, using changes of variable and then the hodograph transform. Along the way, we prove and make use of new Caffarelli-Peral type $W^{1, p}$ estimates for such degenerate equations. Much of our approach appears new even in the case of Alt-Phillips and classical obstacle problems.