Ngô Quốc Anh

April 29, 2011

Uniformly boundedness implies weak convergence

Filed under: Uncategorized — Ngô Quốc Anh @ 5:01

In the study of L^p spaces, there is a well-known theorem that used frequently in PDE. Its statement is the following

Theorem. Let \{f_n\}_n be a sequence of functions in L^p(\Omega) with 1<p<\infty. If there is a positive constant C and a function f such that

\|f_n\|_p \leqslant C, \quad \text{ for any } n

and

f_n \to f \quad \text{ almost everywhere in } \Omega

then f \in L^p(\Omega) and f_n \rightharpoonup f weakly in L^p(\Omega).

For a proof, we refer the reader to Brezis’s book. The idea is as follows

Step 1. If f_n \rightharpoonup f weakly in L^p(\Omega) and f_n \to \overline f almost everywhere in \Omega then f = \overline f almost everywhere.

The idea of the proof of Step 1 is well-known. If f_n \rightharpoonup f weakly in L^p(\Omega), then we can select a new sequence g_n such that g_n \to f strongly in \Omega. Moreover, g_n \to f almost everywhere.

Step 2. The boundedness and the reflexivity of L^p(\Omega)  imply there is a subsequence of \{f_n\}_n converging to f.

This is a standard property.

Step 3. For any subsequence, if we can find a subsubsequence converging to the same limit, the whole sequence also converges to the limit.

This can be proved by contradiction.

The advantage of the theorem is very clear. If we want to prove something like

\displaystyle \int_\Omega (u_n)^pvd\mu \to \int_\Omega u^pvd\mu

as n\to \infty, u_n \in H^1(\Omega), u_n \rightharpoonup u, and for certain p higher enough that we don’t have any compactness property, the uniformly bounded of \{u_n\}_n in some L^q space, q>p, will help us.

April 27, 2011

The Picone identity for p-Laplacian

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 19:37

Last time we discussed the Picone identity for general purpose [here]. Now we present a generalization of this for p-Laplacian operator. This can be seen from the a paper by Walter Allegretto and Yin Xi Huang [here].

Let v > 0, u \geqslant 0 be differentiable over a domain \Omega. Denote

\displaystyle L(u,v) = {\left| {\nabla u} \right|^p} - p{\left( {\frac{u}{v}} \right)^{p = 1}}\nabla u \cdot \nabla v{\left| {\nabla v} \right|^{p - 2}} + (p - 1){\left( {\frac{u}{v}} \right)^p}{\left| {\nabla v} \right|^p}

and

\displaystyle R(u,v) = {\left| {\nabla u} \right|^p} - \nabla \left( {\frac{{{u^p}}}{{{v^{p - 1}}}}} \right) \cdot \nabla v{\left| {\nabla v} \right|^{p - 2}}.

Then

L(u,v)=R(u,v).

Moreover, L(u, v) \geqslant 0, and L(u, v) = 0 a.e. if and only if \nabla \left(\frac{u}{v}\right) = 0 a.e. , i.e. u = kv for some constant k in each component of the domain.

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April 25, 2011

The Monge-Ampère equations: An invariant

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 3:16

Howdy, today, we study a simple invariant concerning the Monge-Ampère equations. Let us consider the following simple case

\displaystyle\det (u_{x_ix_j})=u^p

in \mathbb R^n, where 0<p<n. For a beginner, we refer to an entry from wikipedia [here].

We now look for a function of the form

w(x)=m^\alpha u(mx)

where \alpha is a constant to be determined. A quick calculation shows that

w_{x_i}(x)=m^{\alpha+1} u_{x_i}(mx)

and

w_{x_ix_j}(x)=m^{\alpha+2} u_{x_ix_j}(mx).

Thus

\displaystyle \det (w_{x_ix_j}) = m^{(\alpha+2)n}\det (u_{x_ix_j}(mx))=m^{(\alpha+2)n} u(mx)^p=m^{(\alpha+2)n-p\alpha} w(x)^p.

Therefore, \alpha is chosen so that (\alpha+2)n-p\alpha=0. Hence

\displaystyle\alpha=\frac{2n}{p-n}.

In conclusion,

\displaystyle m^\frac{2n}{p-n} u(mx)

is also a solution. This property can be found in a paper by Kutev [here].

April 22, 2011

On Costa-Hardy-Rellich inequalities


This note is to concern a recent result by David G. Costa [here]. Here the statement

Theorem 1.1. For all a,b\in \mathbb R and u \in C^\infty_0(\mathbb R^N\backslash\{0\}) one has

\displaystyle\left| {\frac{{N - 2 - \gamma }}{2}\int_{\mathbb R^N} {\frac{{|\nabla u{|^2}}}{{|x{|^\gamma }}}dx} + \gamma \int_{\mathbb R^N} {\frac{{{{(x \cdot \nabla u)}^2}}}{{|x{|^{\gamma + 2}}}}dx} } \right| \leqslant {\left( {\int_{\mathbb R^N} {\frac{{|\Delta u{|^2}}}{{|x{|^{2b}}}}dx} } \right)^{\frac{1}{2}}}{\left( {\int_{\mathbb R^N} {\frac{{|\nabla u{|^2}}}{{|x{|^{2a}}}}dx} } \right)^{\frac{1}{2}}}

where \gamma=a+b+1. In addition, if \gamma \leqslant N-2, then

\displaystyle\widehat C\int_{\mathbb R^N} {\frac{{{{(x \cdot \nabla u)}^2}}}{{|x{|^{\gamma + 2}}}}dx} \leqslant {\left( {\int_{\mathbb R^N} {\frac{{|\Delta u{|^2}}}{{|x{|^{2b}}}}dx} } \right)^{\frac{1}{2}}}{\left( {\int_{\mathbb R^N} {\frac{{|\nabla u{|^2}}}{{|x{|^{2a}}}}dx} } \right)^{\frac{1}{2}}}

where the constant \widehat C=|\frac{N+a+b-1}{2}| is sharp.

Here’s the proof.

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April 19, 2011

Flows for Q-curvature problems

Filed under: Riemannian geometry — Ngô Quốc Anh @ 4:14

At the time when I studied the flow method, I was confused whenever we use the evolution for metric or conformal factor. This is why I am writing this top to address this question.

For the convenience, we shall consider the flow for Q-curvature over an n-dimensional sphere \mathbb S^n. Let g be a metric on \mathbb S^n which is conformally equivalent to the standard metric g_0. If we write

g=e^{nw}g_0

then the Paneitz operator with respect to g becomes

P_g = e^{-nw}P_{g_0}

and the Q-curvature of g is given by

Q_g=e^{-nw}(Q_{g_0}+P_{g_0}w).

As the evolution of metric, the Q-curvature flow in defined as

\displaystyle \frac{\partial}{\partial t}g(t)=-(Q_{g(t)}-\overline Q_{g(t)})g(t)

where \overline Q denotes the average of Q, that is

\displaystyle \overline Q_{g(t)}=\frac{1}{\int_{\mathbb S^n} dv_{g(t)}}\int_{\mathbb S^n} Q_{g(t)}dv_{g(t)}.

Suppose g(t) is conformal to g_0, that is

g(t)=e^{2w(t)}g_0

we have

\displaystyle\frac{\partial }{{\partial t}}g(t) = \frac{\partial }{{\partial t}}\left( {{e^{2w(t)}}{g_0}} \right) = 2{e^{2w(t)}}{g_0}\frac{\partial }{{\partial t}}w(t).

Therefore,

\displaystyle 2{e^{2w(t)}}{g_0}\frac{\partial }{{\partial t}}w(t) = \frac{\partial }{{\partial t}}g(t) = - ({Q_{g(t)}} - {{\overline Q}_{g(t)}})g(t) = - ({Q_{g(t)}} - {{\overline Q}_{g(t)}}){e^{2w(t)}}{g_0}

which is equivalent to

\displaystyle \frac{\partial }{{\partial t}}w(t) = - \frac{1}{2}({Q_{g(t)}} - {{\overline Q}_{g(t)}}).

This is the Q-curvature flow written as the evolution of conformal factor w.

April 16, 2011

Pushforward (differential) of a smooth map

Filed under: Riemannian geometry — Ngô Quốc Anh @ 21:09

Suppose that \varphi : M \to N is a smooth map between smooth manifolds; then the differential of \varphi at a point x is, in some sense, the best linear approximation of \varphi near x. It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, it is a linear map from the tangent space of M at x to the tangent space of N at \varphi (x). Hence it can be used to push forward tangent vectors on M to tangent vectors on N.

The differential of a map \varphi is also called, by various authors, the derivative or total derivative of \varphi, and is sometimes itself called the pushforward.

"If a map, φ, carries every point on manifold M to manifold N then the pushforward of φ carries vectors in the tangent space at every point in M to a tangent space at every point in N"

If a map, \varphi, carries every point on manifold M to manifold N then the pushforward of \varphi carries vectors in the tangent space at every point in M to a tangent space at every point in N.

Motivation. Let \varphi :U\to V be a smooth map from an open subset U of \mathbb R^m to an open subset V of \mathbb R^n. For any point x in U, the Jacobian of \varphi at x (with respect to the standard coordinates) is the matrix representation of the total derivative of \varphi at x, which is a linear map

\mathrm d \varphi_x:\mathbb R^m\to\mathbb R^n

from \mathbb R^m to \mathbb R^n. We wish to generalize this to the case that \varphi is a smooth function between any smooth manifolds M and N.

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April 13, 2011

How good the Hardy inequality is?

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 3:34

Before going, let us recall the so-called Hardy inequality from this note below

Theorem (Hardy’s inequality). Let u \in \mathcal D^{1,2}(\mathbb R^n) with n \geqslant 3. Then

\displaystyle\frac{{{u^2}}}{{{{\left| x \right|}^2}}} \in {L^1}({\mathbb{R}^n})

and

\displaystyle {\left( {\frac{{n - 2}}{2}} \right)^2}\int_{{\mathbb{R}^n}} {\frac{{{u^2}}}{{{{\left| x \right|}^2}}}dx} \leqslant \int_{{\mathbb{R}^n}} {{{\left| {\nabla u} \right|}^2}dx}.

The constant {\left( {\frac{{n - 2}}{2}} \right)^2} is the best possible constant.

The purpose of this note is to show that the Hardy integral cannot be improved in the usual sense, that is, there are no nontrivial potential V \geqslant 0 and no exponent q>0 such that, for any function u,

\displaystyle C{\left( {\int_{{\mathbb{R}^n}} {V(x)|u{|^n}dx} } \right)^{\frac{2}{n}}} \leqslant {\left( {\frac{2}{{n - 2}}} \right)^2}\int_{{\mathbb{R}^n}} {{{\left| {\nabla u} \right|}^2}dx} - \int_{{\mathbb{R}^n}} {\frac{{{u^2}}}{{{{\left| x \right|}^2}}}dx}

for some positive constant C.

Indeed, let us construct the following family of test functions u_\varepsilon as follows

\displaystyle {u_\varepsilon }(x) = \begin{cases}|x{|^{ - \frac{{n - 2}}{2} + \varepsilon }},&|x| \leqslant 1,\\ |x{|^{ - \frac{{n - 2}}{2} - \varepsilon }},&|x| > 1.\end{cases}

(more…)

April 10, 2011

A property of the Weingarten map (the sharp operator)

Filed under: Riemannian geometry — Ngô Quốc Anh @ 23:29

Let M be a surface, p \in M and \eta the unit normal vector (field) of M defined along a neighborhood W of p.

Definition (the Weingarten map). The Weingarten map or the sharp operator L: T_pM \to T_pM is defined to be

\displaystyle L(X)=\nabla_X\eta

for any X\in T_pM.

To see this definition is well-defined, in this note, we shall prove that indeed L(X) \in T_pM for any tangent vector X\in T_pM. To this purpose, we have to show that

\langle \nabla_X\eta,\eta \rangle =0.

Be means of the normality, it holds \langle \eta,\eta \rangle =1 which implies

\nabla_X\langle\eta,\eta \rangle =0.

By the property saying that the connection \nabla is compatible with the metric, it holds

\nabla_X\langle\eta,\eta \rangle =\langle\nabla_X\eta,\eta \rangle +\langle\eta,\nabla_X\eta \rangle.

Thus

2\langle\nabla_X\eta,\eta \rangle=0.

The proof is complete.

The role of the Weingarten map is important since it is a tool to introduce the second fundamental form II:T_pM \times T_pM \to \mathbb R given by

II(X,Y)=\langle L(X),Y\rangle .

Apparently, for a plan, the second fundamental form is identically zero and for a sphere S of radius r, its second fundamental form is nothing but \frac{1}{r} {\rm id}.

April 8, 2011

100k hits

Filed under: Linh Tinh — Ngô Quốc Anh @ 20:52

Hello everyone,

This is a great news and I should share this immediately :D!

My blog has just gone through 100,000 hits at this moment :). I take this opportunity to thank all of you for your interest in my blog, for your suggestion, for your beautiful comments, and for everything…

As can be seen, I have started using this wordpress blog since May, 2007. However, there was not much information here until the time I came to Singapore to pursue my PhD degree. That was August, 2008. During this period of time, I have learnt much and of course I have some spare time to write down things that I think it may be useful.

A bit for the future, I will keep writing things that I usually face during my mathematics career. Further than that, it totally depends on how much time I will have. Anyway, please do enjoy what we have here.

Again, thank you and have a nice day :),

Ngo Quoc Anh.

April 7, 2011

Weak comparison principle: p-Laplacian with Neumann boundary condition

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 20:25

I am going to summarize some known comparison principles. I will start with a weak comparison principle for p-Laplacian with Neumann boundary condition. This is adapted from a recent paper by J. Giacomoni et al. published in Differential Integral Equations [here]. I will keep the numbering for the convenience.

Lemma 3.1. Let u,v \in W^{1,N}(\Omega) be non-negative functions satisfying

\displaystyle -\Delta_N u + u^{N-1} \geqslant -\Delta_N v + v^{N-1}

in \Omega and

\displaystyle {\left| {\nabla u} \right|^{N - 2}}\frac{{\partial u}}{{\partial \nu }} \geqslant {\left| {\nabla v} \right|^{N - 2}}\frac{{\partial v}}{{\partial \nu }}

on the boundary \partial \Omega in the strong sense. Then u\geqslant v in \overline\Omega.

Proof. The trick is that if we want to prove u\geqslant v in \overline\Omega then we just show that

(u-v)^- \equiv 0

in \overline\Omega. To this purpose, we use (u-v)^- as a test function in the equation

\displaystyle - ({\Delta _N}u - {\Delta _N}v) + ({u^{N - 1}} - {v^{N - 1}}) \geqslant 0

and integrate to achive

(more…)

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