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

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 \[ f'(u)\in L^q_{\text{loc}}\text{ for every }q<q_f . \] Combined with our $\varepsilon$-regularity theorem, this yields quantitative bounds for the singular set, in particular \[ \dim_{\mathcal H}\Sigma(u)\le n-2q_f. \] Remarkably, our exponent $q_f$ recovers the sharp thresholds for all standard model nonlinearities, including $f(t)=(1+t)^p$, $e^t$, and $(1-t)^{-p}$. Also, this result provides the first general quantitative singular-set estimates for stable semilinear equations beyond the model nonlinearities. Finally, in the two-dimensional case, we provide a complete picture by proving the universal Hessian estimate \[ \|D^2u\|_{L^\infty(B_{1/2})}\le C\|u\|_{L^1(B_1)}, \] where $C$ depends neither on $u$ nor on $f$. This $C^{1,1}$ regularity is essentially optimal: one cannot expect $C^{2,\alpha}$ estimates for any $\alpha>0$, and in general even $C^2$ regularity should fail.