Stable Semilinear Elliptic Equations: $\epsilon$-Regularity à la Brezis and Dimensional Bounds for the Singular Set

Alessio Figalli, Federico Franceschini

We develop a quantitative partial regularity theory for stable solutions of \[ -Δu=f(u), \] where $f:\mathbb{R}\to [0,+\infty ]$ is increasing and convex. The theory is uniform in t…