About this Event
182 MEMORIAL DR, Cambridge, MA 02139
https://math.mit.edu/pde-analysis/Speakers: Federico Franceschini (Institute for Advanced Study)
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 −∆u = f (u) can be singular (i.e., un-
bounded) if the dimension n ≥ 10.
Brezis conjectured that if x = 0 is such a singular point, then
f ′(u(x)) blows-up like |x|2−n. Villegas showed that such a strong
statement fails for general nonlinearities.
In this talk, we prove — for all nonlinearities — a version of
Brezis conjecture, which is essentially the best one can obtain in
view of the counterexamples of Villegas. Building on this result we
then show that the singular set has dimension n-10, at least for a
large class of nonlinearities that includes the most relevant cases.
This is a joint work with Alessio Figalli.