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

We develop a quantitative partial regularity theory for stable solutions of $$-\mathrm{\Delta }u=f(u),$$ where $f:\mathbb{R}\to [0,+\mathrm{\infty }]$ is increasing and convex. The theory is uniform in the nonlinearity and allows for a finite or infinite blow-up level $T_f\in (-\mathrm{\infty },+\mathrm{\infty }]\mathrm{.}$ Our first result is a universal $\epsilon$-regularity criterion that answers a celebrated question of Brezis: smallness of the scale-invariant mass of the stability potential $f^{\mathrm{\prime }}(u)$ forces Hölder regularity. Moreover, if $T_f<+\mathrm{\infty }$, 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 $q_f\ge 1$, given explicitly in terms of the asymptotic behavior of $f$, $f^{\mathrm{\prime }}$, and $f^{\mathrm{\prime }\mathrm{\prime }}$, such that $$f^{\mathrm{\prime }}(u)\in L_{\text{loc}}^q\text{ for every }q<q_f\mathrm{.}$$ Combined with our $\epsilon$-regularity theorem, this yields quantitative bounds for the singular set, in particular $${\dim }_{\mathcal{H}}\mathrm{\Sigma }(u)\le n-2q_f\mathrm{.}$$ 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 $$\mathrm{\parallel }{D}^{2}u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(B_{1\mathrm{/}2})}\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{L^1(B_1)},$$ 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.