Stable solutions to semilinear elliptic equations

Stable 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 answers also a famous open problem posed
by Brezis, concerning the regularity of extremal solutions to the Gelfand
problem.

https://www.youtube.com/channel/UChNFQvPQRp6G5CDPmTIKshw