The free boundary for semilinear problems with highly oscillating singular terms

We investigate general semilinear (obstacle-like) problems of the form $\Delta u=f\left(u\right)$, where $f\left(u\right)$ 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 partial differential equations (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.