# Stability for Faber-Krahn inequalities and the ACF formula

# Stability for Faber-Krahn inequalities and the ACF formula

**Zoom Link: https://princeton.zoom.us/j/95636729444**

**We will also use a Gather.town platform for chats before and after the talks. https://gather.town/app/Ybqg2gluvOoZuIvq/JointSeminar**

The Faber-Krahn inequality states that the first Dirichlet eigenvalue of the Laplacian on a domain is greater than or equal to that of a ball of the same volume (and if equality holds, then the domain is a translate of a ball). Similar inequalities are available on other manifolds where balls minimize perimeter over sets of a given volume. I will discuss the stability problem for such inequalities: if the eigenvalue of a set is close to a ball, how similar to a ball must the set look like? I will also explain an application of sufficiently strong stability results to quantifying the behavior of the Alt-Caffarelli-Friedman monotonicity formula, which has implications for free boundary problems with multiple phases. This is based on recent joint work with Mark Allen and Robin Neumayer.