# Ngô Quốc Anh

## January 6, 2011

### The Alexandrov-Bol inequality

Filed under: Giải Tích 6 (MA5205) — Tags: , — Ngô Quốc Anh @ 19:55

In the literature, there is an inequality called the Alexandrov-Bol inequality which is frequently used in partial differential equations. Here we just recall its statement without any proof.

Theorem. Let $\Omega$ be a good domain in $\mathbb R^2$. Assume $p \in C^2(\Omega)\cap C^0(\overline \Omega)$ be a positive function satisfying the elliptic inequality

$\displaystyle -\Delta \log p \leqslant p$

in $\Omega$. Then it holds

$\displaystyle l^2(\partial\Omega) \geqslant \frac{1}{2} \big(8\pi-m(\Omega)\big)m(\Omega)$

where

$\displaystyle l(\partial\Omega)=\int_{\partial\Omega}\sqrt{p}ds$

and

$\displaystyle m(\Omega)=\int_\Omega pdx$.

An analytic proof was given by C. Bandle aroud 1975 when she assumed $p$ to be real analytic. The above version was due to Suzuki in an elegant paper published in the Ann. Inst. H. Poincare in 1992 [here]. The proof is mainly depended on the isoperimetric inequality for the flat Riemannian surfaces. We refer the reader to the paper by Suzuki for the proof.

do you know if it exists a Bol type inequality for the $N$-laplacian?
• Hi Fab, I do not know, however, do we have a similar generalization for $N$-Laplacian?