## ENSPM2021 Talk

**Title:** Stable and finite Morse index solutions to semilinear elliptic equations

**Abstract:** Stable and finite Morse index solutions to semilinear elliptic PDEs appear in several problems. It is known since the 1970’s that, in dimension n > 9, there exist singular stable solutions. In this talk, I will describe a recent work with Cabré, Ros-Oton, and Serra, where we prove that stable solutions in dimension n ≤ 9 are smooth. This also answers a famous open problem posed by Brezis, concerning the regularity of extremal solutions to the Gelfand problem. Also, I will discuss a recent analog result with Zhang for finite Morse index solutions.

## Alessio Figalli (ETH Zurich, Switzerland)

He is an Italian mathematician working at ETH Zurich in the broad areas of Calculus of Variations and Partial Differential Equations, with particular emphasis on optimal transportation, Monge-Ampère equations, functional and geometric inequalities, elliptic PDEs of local and non-local type, free boundary problems, Hamilton-Jacobi equations, transport equations with rough vector-fields, and random matrix theory. He was awarded the Prix and Cours Peccot in 2012, the EMS Prize in 2012, the Stampacchia Medal in 2015, the Feltrinelli Prize in 2017, he was awarded a European Research Council (ERC) grant in 2016, and he won the Fields Medal in 2018 “for his contributions to the theory of optimal transport, and its application to partial differential equations, metric geometry, and probability”. He was an invited speaker at the International Congress of Mathematicians 2014.