PDE Seminar: abstract

Abstract

Let $Ω \subset ℝ^{n}$ be a convex domain and let $f : Ω \to ℝ$ be a positive, subharmonic function (i.e. $Δ f \geq 0$ ). Then

\frac{1}{| Ω |} \int_{Ω} f d x \leq \frac{c_{n}}{| \partial Ω |} \int_{\partial Ω} f d σ,

where $c_{n} \leq 2 n^{3 ∕ 2}$ . This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_{n} \geq n - 1$ . As a byproduct, we establish the following sharp geometric inequality for two convex domains where one contains the other $Ω_{2} \subset Ω_{1} \subset ℝ^{n}$ :

\frac{| \partial Ω_{1} |}{| Ω_{1} |} \frac{| Ω_{2} |}{| \partial Ω_{2} |} \leq n .

On the Hermite-Hadamard formula in higher dimensions

Abstract