Stable solutions to reaction-diffusion elliptic problems

We are concerned with stable solutions to reaction-diffusion elliptic PDEs. We begin with regularity questions, first addressing the classical Laplacian. In joint work with Figalli, Ros-Oton, and Serra, we proved that stable solutions are smooth up to the optimal dimension 9, thereby solving an open problem posed by Brezis in the mid-1990s. We describe this result and also discuss related progress and open problems for the fractional Laplacian -- arising naturally in boundary reaction problems -- , the $p$-Laplacian, and minimal surfaces. We then turn to existence questions, starting with the Casten-Holland and Matano theorem for interior reactions, which states that no nonconstant stable solution exists in convex domains under zero Neumann boundary conditions. We present a recent result with Consul and Kurzke (forthcoming) establishing that the analogous statement fails for boundary reactions. This requires the development of a new Ginzburg-Landau theory for real-valued functions and the analysis of the half-Laplacian on the real line, for which we present new results and open problems.