Stable Semilinear Elliptic Equations: $\varepsilon$-Regularity \`a la Brezis and Dimensional Bounds for the Singular Set

Abstract

We develop a quantitative partial regularity theory for stable solutions of \[ -\Delta u=f(u), \] where $f:\mathbb R \to [0,+\infty]$ is increasing and convex. The theory is uniform in the nonlinearity and allows for a finite or infinite blow-up level $T_f\in(-\infty,+\infty].$ Our first result is a universal $\varepsilon$-regularity criterion that answers a celebrated question of Brezis: smallness of the scale-invariant mass of the stability potential $f'(u)$ forces H\"older regularity. Moreover, if $T_f<+\infty$, the same smallness condition forces almost quadratic contact between the solution and the blow-up level $T_f$. This result is optimal and, in particular, covers the case of MEMS-type nonlinearities. Our second result identifies a critical exponent $q_f\ge1$, given explicitly in terms of the asymptotic behavior of $f$, $f'$, and $f''$, such that \[ f'(u)\in L^q_{\text{loc}}\text{ for every }q 0$, and in general even $C^2$ regularity should fail.