The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's.
In dimensions 10 and higher, there exist singular stable energy solutions.
In this talk I will describe a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems.

In the framework of a model for the quasistatic crack growth in elasto-plastic homogeneous materials in the planar case, we study the properties of the length of the crack as a function of time.
We prove that, under suitable technical assumptions on the crack path, this monotone function is a pure jump function. Under stronger assumptions we prove also that the number of jumps is finite.

A variational model in the context of the gradient theory for fluid-fluid phase transitions with small scale heterogeneities is studied. In the case where the scale of the small heterogeneities is of the same order of the scale governing the phase transition, the interaction between homogenization and the phase transitions process leads to an anisotropic interfacial energy.
Bounds on the homogenized surface tension are established. In addition, a characterization of the large-scale limiting behavior of viscosity solutions to non-degenerate and periodic Eikonal equations in half-spaces is given.
This is joint work with Riccardo Cristoferi (Radboud University, The Netherlands), Adrian Hagerty (USA), Cristina Popovici (USA), Rustum Choksi (McGill, Canada), Jessica Lin (McGill, Canada), and Raghavendra Venkatraman (NYU, USA).