PDE Seminar Abstracts

On the Hermite-Hadamard formula in higher dimensions

Barbara Brandolini
Universit√† degli Studi di Napoli “Federico II”, Italy
Mon 19th Aug 2019, 12-1pm, Carslaw Room 829 (AGR)


Let Ω n be a convex domain and let f : Ω be a positive, subharmonic function (i.e. Δf 0). Then

1 |Ω|Ωfdx cn |Ω|Ωfdσ,

where cn 2n32. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies cn n - 1. As a byproduct, we establish the following sharp geometric inequality for two convex domains where one contains the other Ω2 Ω1 n:

|Ω1| |Ω1| |Ω2| |Ω2| n.