Ngô Quốc Anh

May 20, 2013

Super polyharmonic property of solutions: The case of single equations

Filed under: PDEs — Ngô Quốc Anh @ 9:39

Recently, I have read a paper by Chen and Li published in the journal CPAA [here or here] about the super polyharmonic property of solutions for some partial differential systems.

In this notes, we consider their result in a very particular case – the single equations. We shall prove the following.

Theorem 2.1. Let p be a positive integer and q>1. For each positive solution u of

\displaystyle (-\Delta)^p u \geqslant u^q

in \mathbb R^n, there holds

\displaystyle (-\Delta)^i u >0 \quad i=\overline{1,p-1}.

Proof. For simplicity, we write

\displaystyle v_i = (-\Delta)^i u \quad i=\overline{1,p-1}.

We must show that v_i >0 for all i. For simplicity, we divide the proof into two steps.

Step 1. Proving v_{p-1}>0.

Assume the contradiction, we then have two possible cases

Case 1. There is some x^0 \in \mathbb R^n such that v_{p-1}(x^0)<0.

Case 2. v_{p-1} \geqslant 0 and there is a point x_0 such that v_{p-1}(x_0)=0.

(more…)

April 25, 2013

The Cauchy formula for repeated integration

Filed under: Các Bài Tập Nhỏ, Giải Tích 2 — Ngô Quốc Anh @ 23:41

The Cauchy formula for repeated integration, named after Augustin Louis Cauchy, allows one to compress n antidifferentiations of a function into a single integral.

Let f be a continuous function on the real line. Then the n-th repeated integral of f based at a,

\displaystyle f^{(-n)}(x) = \int_a^x \int_a^{\sigma_1} \cdots \int_a^{\sigma_{n-1}} f(\sigma_{n}) \, \mathrm{d}\sigma_{n} \cdots \, \mathrm{d}\sigma_2 \, \mathrm{d}\sigma_1,

is given by single integration

\displaystyle f^{(-n)}(x) = \frac{1}{(n-1)!} \int_a^x\left(x-t\right)^{n-1} f(t)\,\mathrm{d}t.

A proof is given by induction. Since f is continuous, the base case follows from the Fundamental theorem of calculus

\displaystyle\frac{\mathrm{d}}{\mathrm{d}x} f^{(-1)}(x) = \frac{\mathrm{d}}{\mathrm{d}x}\int_a^x f(t)\,\mathrm{d}t = f(x);

where

\displaystyle f^{(-1)}(a) = \int_a^a f(t)\,\mathrm{d}t = 0.

(more…)

April 18, 2013

A lower bound for solutions involving distance functions

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 21:14

In this note, we discuss an useful lemma and its beautiful proof given by Brezis and Cabré in a paper published in Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. in 1998. The full article can be freely downloaded from here. Before saying anything further, let us state the lemma.

Lemma 3.2. Suppose that h \geq 0 belongs to L^\infty (\Omega). Let v be the solution of

\left\{\begin{array}{rcl} - \Delta v &=& h \quad \text{ in }\Omega , \hfill \\ v &=& 0 \quad \text{ on }\partial \Omega . \hfill \\ \end{array}\right.

Then

\displaystyle\frac{{v(x)}}{{\text{dist}(x,\partial \Omega )}} \geqslant c\int_\Omega {h\text{dist}(x,\partial \Omega )} ,\qquad\forall x \in \Omega,

where c>0 is a constant depending only on \Omega.

This type of estimate frequently uses in the literature. We now show the proof of the lemma.

(more…)

April 1, 2013

A proof of the uniqueness of solutions of the Lichnerowicz equations in the compact case

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 4:38

In this small note, we aim to derive some uniqueness property of solutions of the following PDE

\displaystyle -a\Delta_g u +\text{Scal}_g u =|\sigma|_g^2 u ^{-2\kappa-3}-b\tau^2 u^{2\kappa+1}

on a compact manifold (M,g) where \kappa=\frac{2}{n-2}, a=2\kappa+4, and b=\frac{n-1}{n}.

We assume that \phi_1 and \phi_2 are solutions of the above PDE. Setting \phi=\frac{\phi_2}{\phi_1}. We wish to prove that \phi=1.

Let us consider the following trick basically due to David Maxwell, see this paper. Let \widetilde g= \phi_1^{2\kappa}g. Then the well-known formula for the Laplace-Beltrami operator \Delta_g, which is

\displaystyle {\Delta _g}u = \frac{1}{{\sqrt {\det g} }}{\partial _i}(\sqrt {\det g} {g^{ij}}{\partial _j}u)

helps us to write

\displaystyle {\Delta _{\widetilde g}}u = \phi _1^{ - 2\kappa } \Big({\Delta _g}u + \frac{2}{{{\phi _1}}}{\nabla _g}{\phi _1}{\nabla _g}u \Big).

Consequently,

\displaystyle {\Delta _g}\phi = \phi _1^{2\kappa }{\Delta _{\widetilde g}}\phi - \frac{2}{{{\phi _1}}}{\nabla _g}{\phi _1}{\nabla _g}\phi .

We now calculate - a{\Delta _g}{\phi _2} + {R_g}{\phi _2} as follows.

(more…)

March 12, 2013

The generalized maximum principle

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 9:09

Let us start our notes with a very fundamental maximum principle for any strongly second order elliptic operator. We have

Theorem (Maximum principle). Let u satisfy the differential inequality

\displaystyle L[u] = \sum\limits_{i,j = 1}^n {{a_{ij}}(x)\frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}}} + \sum\limits_{k = 1}^n {{b_k}(x)\frac{{\partial u}}{{\partial {x_k}}}} \geqslant 0

in a domain D where L is uniformly elliptic. Suppose the coefficients a_{ij} and b_k are uniformly bounded. If u attains a maximum M at a point of D, then u = M in D.

In order to memorize the above result, let us think about the parabola y=x^2 with x\in[-1,1] and L=\Delta. In this one-dimentional case, L[y]=2 \geqslant 0 which confirms that y only achieves its maximum at x=\pm 1.

As you may know the operator L is only assumed to be strongly elliptic which only effects the coefficients a_{ij}. Regarding to the coefficients b_k, we only assume these are uniformly bounded. However, the uniform ellipticity of the operator L and the boundedness of the coefficients are not really essential as you can check in the proof. Besides, the domain D need not be bounded in this version.

Now for operators of the form (L + h), we still have a result analogous to the above.

Theorem (Maximum principle). Let u satisfy the differential inequality

\displaystyle (L + h)[u] >0

with h <0, with L uniformly elliptic in D, and with the coefficients of L and h bounded. If u attains a non-negative maximum M at an interior point of D, then u = M.

Clearly, the assumption h<0 and M \geqslant 0 are crucial as we may face some difficulty as raised in this note. Counterexamples are easily obtained if h > O. For example, the function u = \exp(-r^2) has an absolute maximum at r= 0.

(more…)

March 9, 2013

On the integrability of the inverse of functions in the Sobolev space H_0^1

Filed under: PDEs — Ngô Quốc Anh @ 9:05

Yesterday, I read a recent paper by S. Yijing and Z. Duanzhi published in the journal Calculus of Variations in 2013. In one of their results, they proved the following

Theorem 2. Let \Omega \subset\mathbb R^N, N \geq 3 be bounded open set with smooth boundary. If p\geq 3, then

\displaystyle \int_\Omega |u|^{1-p}dx =+\infty, \quad \forall u \in H_0^1(\Omega).

Since p \geq 3, it is clear that 1-p \leq -2. By definition, \int_\Omega |u|^2dx <+\infty which gives us some comparison between u and its inverse power.

In order to prove the above theorem, the authors first proved the following

Theorem 1. Let \Omega \subset\mathbb R^N, N \geq 3 be bounded open set with smooth boundary, the function h\in L^1(\Omega) is positive a.e. in \Omega, and p>1. Then the equation

\displaystyle\begin{array}{rcl} \Delta u + h(x){u^{ - p}} &=& 0 \quad \text{ in } \Omega \hfill \\ u &>& 0 \quad \text{ in } \Omega \\ u &=& 0 \quad \text{ on }\partial \Omega \end{array}

admits a unique H_0^1-solution if and only if there exists u_0 \in H_0^1(\Omega) such that

\displaystyle \int_\Omega h(x) |u_0|^{1-p}dx <+\infty.

The proof of Theorem 1 is variational which makes use of the Nehari manifold and the fibering map. Basically, the authors tried to minimize the functional

(more…)

March 1, 2013

PhD Thesis: The Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds

Filed under: Luận Văn — Ngô Quốc Anh @ 6:12

Eventually, my PhD thesis had been released worldwide :) .

Title: The Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds
Authors: NGO QUOC ANH
Supervisor: XU XINGWANG
Keywords: Einstein-scalar field equation, Lichnerowicz equation, Critical exponent, Negative exponent, Conformal method, Variational method
Issue Date: 2012
Abstract: We establish some new existence and multiplicity results for positive solutions of the following Einstein-scalar field Lichnerowicz equations on compact manifolds (M,g) without the boundary of dimension n \geqslant 3,

\displaystyle -\Delta_g u + hu = fu^\frac{n+2}{n-2} + au^{-\frac{3n-2}{n-2}},

with either a negative, a zero, or a positive Yamabe-scalar field conformal invariant h. These equations arise from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity. The variational method can be naturally adopted to the analysis of the Hamiltonian constraint equations. However, it arises analytical difficulty, especially in the case when the prescribed scalar curvature-scalar field function f may change sign. To our knowledge, such a problem in its most generic case remains open. Finally, we establish some Liouville type results for a wider class of equations with constant coefficients including the Einstein-scalar field Lichnerowicz equation with constant coefficients.
Department: MATHEMATICS
Degree Conferred: DOCTOR OF PHILOSOPHY
Document Type: Thesis

If you are interested in my thesis, you can freely download it from the link http://scholarbank.nus.edu.sg/handle/10635/36344.

Thank you.

February 10, 2013

2013 Tết Holiday

Filed under: Linh Tinh — Ngô Quốc Anh @ 14:24

2013 Tet Holiday

Tết Holiday

Vietnamese New Year, more commonly known by its shortened name Tết or Tết Nguyên Đán, is the most important and popular holiday and festival in Vietnam. It is the Vietnamese New Year marking the arrival of spring based on the Chinese calendar, a lunisolar calendar. For those who do not know about Tết, please read an article in wikipedia for details.

At the first moment of the new year, I wish you a good health and prosperity all year round and thank you for your interest in my blog.

January 29, 2013

Say HI in MuPAD

Filed under: Linh Tinh — Tags: — Ngô Quốc Anh @ 5:37

I found this interesting note regarding to Matlab. In that note, they plotted the word HI using Matlab. Here I try to use MuPAD in order to get a slightly better picture.

As mentioned in the note, the full function we need to use is

\displaystyle e^{-x^2-\frac{1}{2}y^2} \cos(4x) + e^{-3\big( (x+\frac{1}{2})^2+\frac{1}{2}y^2 \big)}.

If you plot that full function, what you are going to have is the following picture

HI 0

(more…)

January 22, 2013

PhD Thesis Defense

Filed under: Linh Tinh, Luận Văn — Ngô Quốc Anh @ 15:46

I just passed my PhD defense on 18 Jan, 2013. Following is the front page of the slides I used during the defense.

The committee of my defense consists of

Since the thesis contains some unpublished results, I cannot provide the slides here but you can email me if interested.

Title of my PhD thesis defense

Title page of the slides used in my PhD thesis defense

Older Posts »

Theme: Shocking Blue Green. Blog at WordPress.com.

Follow

Get every new post delivered to your Inbox.

Join 28 other followers