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

Abstract

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 the nonlinearity and allows for a finite or infinite blow-up level $T_f\in (-\infty ,+\infty ].$ 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 $f^{\prime }(u)$ forces Hölder 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 $q_f\ge 1$, given explicitly in terms of the asymptotic behavior of $f$, $f^{\prime }$, and $f^{\prime \prime }$, such that \[ f'(u)\in L^q_{\text{loc}}\text{ for every }q<q_f . \] Combined with our $\epsilon$-regularity theorem, this yields quantitative bounds for the singular set, in particular \[ \dim_{\mathcal H}Σ(u)\le n-2q_f. \] 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.

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