The free boundary for semilinear problems with highly oscillating singular terms

TL;DR

This paper studies the regularity of free boundaries in semilinear obstacle problems with highly oscillatory singularities, establishing smoothness at flat points through novel analytical techniques.

Contribution

It introduces new methods to prove the $C^ abla$ regularity of free boundaries in complex oscillatory semilinear problems, extending classical obstacle problem results.

Findings

Proved $C^ abla$ regularity of free boundaries at flat points.

Developed new $W^{1,p}$ estimates for degenerate PDEs.

Established Lipschitz continuity of flat free boundaries.

Abstract

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.