Regularity of stable solutions to elliptic equations

In this talk, we address the regularity problem for stable solutions to elliptic equations. Here, a solution is "stable" if the the principal eigenvalue of the linearized equation is nonnegative. In particular, for variational problems, stability amounts to the nonnegativity of the second variation and hence it includes the class of minimizers. The smoothness of stable solutions turns out to be a delicate question which depends on the dimension of the space. For instance, given a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ and a function $f\in C^1(\mathbb{R})$, the semilinear problem $$\{\begin{array}{cl}-\mathrm{\Delta }u=f(u)& \text{ in }\mathrm{\Omega }\\ u=0& \text{ on }\partial \mathrm{\Omega },\end{array}$$ may admit singular stable solutions when $n\ge 10$. For $n\le 9$, it was recently shown that stable solutions are smooth for \emph{all} (nonnegative, nondecreasing, and convex) nonlinearities. The main goal of the talk will be to discuss some extensions of this optimal result to more general operators, including non-variational problems. We will emphasize the precise regularity assumptions needed on the coefficients and the domain.