The APDE seminar on Monday, 2/9, will be given by Federico Franceschini (Stanford) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: The dimension and behaviour of singularities of stable solutions to semilinear elliptic equations

Abstract: Let f(t) be a convex, positive, increasing nonlinearity. It is known that stable solutions of -\Delta u =f(u) can be singular (i.e., unbounded) if the dimension n>9.

Brezis asked wether, if x=0 is such a singular point, then in general f'(u(x)) blows-up like ~|x|^{2-n}, as it happens in the model cases f(u)=u^p, f(u)=e^u.

In this talk I will show the answer to this question and the interesting consequences it entails. This is a joint work with Alessio Figalli.