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).

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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.

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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})}.

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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.

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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.

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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.

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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

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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.

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