# Ngô Quốc Anh

## January 31, 2011

### Stereographic projection, 2

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 19:15

This is a sequel to this topic where we have recalled several properties of the stereographic projection $\pi : \mathbb S^n \to \mathbb R^n$. Recall that by the following transformation

$\displaystyle v(x)=u(\pi^{-1}(x))\left( \frac{2}{1+|x|^2}\right)^\frac{n-2}{2}, \quad x \in \mathbb R^n$

we know that

$\displaystyle -\Delta_g u(\xi) + \frac{n(n-2)}{4}u(\xi) = K(\xi)u(\xi)^\frac{n+2}{n-2} \quad \text{ on } \mathbb S^n$

and

$\displaystyle -\Delta v(x) = K(\pi^{-1}(x))v(x)^\frac{n+2}{n-2} \quad \text{ on } \mathbb R^n$

are equivalent in the weak sense.

The way to see it comes from the following identities

$\displaystyle \int_{{\mathbb{S}^n}} {|\nabla u(\xi ){|^2} + \frac{{n(n - 2)}}{4}u{{(\xi )}^2} = \int_{{\mathbb{R}^n}} {|\nabla v(x){|^2}} }$

and

$\displaystyle \int_{{\mathbb{S}^n}} {|u(\xi ){|^{\frac{{2n}}{{n - 2}}}} = \int_{{\mathbb{R}^n}} {|v(x){|^{\frac{{2n}}{{n - 2}}}}} }$

where $u \in H^1(\mathbb S^n)$.

## January 27, 2011

### Prescribed Q-curvature on 4-manifolds

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 3:40

Let $(M,g)$ be a compact Riemannian $4$-manifold, and let ${\rm Ric}$ and $R$ denote the Ricci tensor and the scalar curvature of $g$, respectively.

A natural conformally invariant in dimension four is

$\displaystyle Q=Q_g=-\frac{1}{6}(\Delta R - R^2 +3|{\rm Ric}|^2)$.

This $Q$ is commonly refered to the $Q$-curvature of metric $g$. The term

$R^2 -3|{\rm Ric}|^2$

is commonly denoted by $6\sigma_2(A)$ where

$\displaystyle A={\rm Ric}-\frac{1}{6}Rg$

the Schouten tensor of $g$ and

$\displaystyle \sigma_2(\cdot)=\frac{1}{2}(\rm tr \; \cdot)^2-\frac{1}{2}|\cdot|^2$

the second elementary symmetric polynomial in its eigenvalues.

## January 23, 2011

### Equivalent forms of the mean field equations

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 1:36

Let $(M,g)$ be a compact Riemannian surface, $h$ be a positive $C^1$ function on $M$. The standard mean field equation can be stated as follows

$\displaystyle {\Delta _g}w + \rho \left( {\frac{{h(x){e^w}}}{{\displaystyle\int_M {h(x){e^w}} }} - 1} \right) = 4\pi \sum\limits_{j = 1}^m {{\alpha _j}({\delta _{{p_j}}} - 1)}$

in $M$, where $p_j \in M$ are given distinct points, $\alpha_j >0$ and $\delta_j$ denotes the Dirac measure with pole at $p_j$. Here the area of $M$ is assumed to be constant $1$ and $\Delta_g$ stands for the Laplace Beltrami operator with respect to $g$.

Clearly, the above PDE is invariant under adding a constant. Hence, $w$ is normalized to satisfy

$\displaystyle \int_M w=0$.

Let $G(x,p)$ be the Green function with pole at $p$, that is,

$\displaystyle\begin{cases}-\Delta_g G(x,p)=\delta_p-1,&{\rm in}\; M,\\\displaystyle\int_M G(x,p)=0,\end{cases}$

and let

$\displaystyle u(x) = w(x) + 4\pi \sum\limits_{j = 1}^m {{\alpha _j}G(x,{p_j})}$.

## January 20, 2011

### On an algebraic identity araising in the Toda systems SU(3)

Filed under: Uncategorized — Ngô Quốc Anh @ 21:42

Recently, in their paper published in J. Diffierential Equations [here], H. Ohtsuka and T. Suzuki prove the following

Suppose we have real numbers a and b satisfying

$a^2-ab+b^2=4\pi(a+b)$.

If we assume

$\min\{a,b\} \geqslant 4\pi$,

then we can prove

$\max\{a,b\}\geqslant 8\pi$.

Furthermore, if we know that both

$2a-b>4\pi,\quad 2b-a>4\pi$,

then we can further show that

$\max\{a,b\}\leqslant 4\left( 1+\frac{2}{\sqrt{3}}\right)\pi \approx 8.61\pi$.

## January 15, 2011

### The Payne’s Maximum Principles

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 15:34

This note, completely based on the elegant paper of L.E. Payne [here], deals primarily with maximum principles for solutions of second order and fourth order elliptic equations. However, some of the results hold for arbitrary sufficiently smooth functions.

Throughout we assume $D$ to be a bounded domain in $\mathbb R^n$ with sufficiently smooth boundary $\partial D$ so that when necessary the governing differential equation will be satisfied on the boundary. In some of the applications we will need $\partial D$ to be a $C^{4+\alpha}$ surface but in most cases this excessive differentiality can be dispensed with.

We shall adopt the summation convention in which repeated Latin indices are to be summed from $1$ to $n$, and we shall use the comma to denote differentiation. The symbol $\partial/\partial \nu$ will be used for the normal derivative directed outward from $D$ on $\partial D$.

Following is the results

1. Inequalities based on the geometry of $D$

We start with the maximum value of the gradient of a function whose normal derivative vanishes on a portion of $\partial D$.

Theorem I. Let $u \in C^2(\overline D)$ have vanishing normal derivative on a portion $\Gamma$ of $\partial D$. Then if the Gaussian curvature o$\Gamma$ is everywhere positive the maximum value of $|{\rm grad} u|^2$ can occur on $\Gamma$ if and only if $u\equiv {\rm constant}$ in $D$.

We also have the following result

Theorem II. Let $u \in C^2(\overline D)$ vanish on a portion $\Gamma_1$ of $\partial D$. Then if the average curvature $K$ is positive at every point of $\Gamma_1$ the maximum value of

$|{\rm grad} u|^2 -2u\Delta u$

cannot occur on $\Gamma_1$ if $u\not \equiv 0$ in $D$.

## January 11, 2011

### An example of sequence of blow-up solutions with finite limiting mass

Filed under: Nghiên Cứu Khoa Học, PDEs — Ngô Quốc Anh @ 14:35

In this note, we recall an example adapted from an elegant paper due to Y.Y. Li and I. Shafrir published in the Indiana Univ. Math. J. in 1994 [here].

Let us consider the asymptitic behavior of sequences of solutions of

$-\Delta u_n=V_n(x)e^{u_n}$

on a bounded domain $\Omega \subset \mathbb R^2$ with $V_n$ a non-negative continuous function. For each solution $u_n$, we call

$\displaystyle \alpha_n := \int_{B_R}V_ne^{u_n}dx$

the mass of $u_n$ (over a ball $B_R$). The terminology limiting mass will be referred to the limit $\lim_{n \to \infty} \alpha_n$.

For simplicity, we assume $V_n \equiv 1$. Given $m$, we are going to construct a sequence of solutions $\{u_n\}$ which blows up exactly at $m$ points, say at $a_1,...,a_m \in D$ where $D$ the unit disc of $\mathbb C$. Our equation reads as

$-\Delta u=e^{u}$

in $D$. Using the Liouville formula for solutions of the above equation, we get

$\displaystyle u(z) = \log \frac{{8|f'(z){|^2}}}{{{{(1 + |f(z){|^2})}^2}}}$

with $f$ an holomorphic function such that $f'(z) \ne 0$.

## January 8, 2011

### A funny limit involving sine function

Filed under: Các Bài Tập Nhỏ, Giải Tích 1 — Ngô Quốc Anh @ 2:32

Today, I have been asked to calculate the following limit

$\displaystyle \mathop {\lim }\limits_{n \to + \infty } \sin (\sin \overbrace {(...(}^n\sin x)...))$

for each fixed $x \in [0,2\pi]$. From the mathematical point of view, we can assume $x \in (-\frac{\pi}{2}, \frac{\pi}{2})$ as we just replace $x$ by $\sin (\sin x))$ if necessary.

There are three possible cases

Case 1. $x \in (0, \frac{\pi}{2})$. In this case, it is well known that function $\frac{\sin x}{x}$ is monotone decreasing since

$\displaystyle {\left( {\frac{{\sin x}}{x}} \right)^\prime } = \frac{{x\cos x - \sin x}}{{{x^2}}} = \frac{{\cos x}}{{{x^2}}}\left( {x - \tan x} \right) \leqslant 0$

in its domain. Consequently, it holds

## 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.