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Nonlinear Analysis

Rectifiability of interfaces with positive Alt-Caffarelli-Friedman limit via quantitative stability

Dennis Kriventsov, Rutgers University

Location:  Hill Center Room 705
Date & time: Tuesday, 21 February 2023 at 1:40PM - 2:40PM

Abstract: The Alt-Caffarelli-Friedman (ACF) monotonicity formula captures fine behavior of pairs of nonnegative harmonic functions which vanish on the mutual boundary of complementary domains. I will describe the following theorem: the set of points with positive limit for the ACF formula, corresponding roughly to where both functions have linear growth away from the interface, is countably n?1 rectifiable. The proof leverages a new quantitative stability property for the ACF formula, which in turn is based on a new quantitative stability result for the Faber-Krahn inequality. Indeed, we show that if ?(?) is the first Dirichlet eigenvalue of a domain ? of volume one and u? is the first eigenfunction, then |u_? ? u_B| 2 ? C[?(?) ? ?(B)] for some ball B with |B| = |?| = 1. Our proof of this relies on regularity theory for some "critical" modifications of Bernoulli-type free boundary problems. This is based on joint work with Mark Allen and Robin Neumayer.

Seminar website:  Nonlinear Analysis Seminar (https://sites.math.rutgers.edu/~yyli/NonlinearAnalysisSeminar.html) 

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