# Ngô Quốc Anh

## 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.

## 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.

## September 14, 2010

### Asympotic behavior of integrals, 2

Filed under: Giải Tích 6 (MA5205), PDEs — Tags: — Ngô Quốc Anh @ 15:07

We now prove the following result

Theorem. Let $u$ and $f$ be two smooth functions on $\mathbb R^2$ satisfying

$\Delta u(x)=f(x), \quad x \in \mathbb R^2$.

Suppose that $f$ is bounded and also $f \in L^1(\mathbb R^2)$ and

$|u(x)| \leqslant o(|x|), \quad |x| \to \infty$.

Then

$\displaystyle\mathop {\lim }\limits_{|x| \to \infty } \frac{{u(x)}}{{\log |x|}} = \frac{1}{{2\pi }}\int_{{\mathbb{R}^2}} {f(y)dy}$.

## September 7, 2010

### Asympotic behavior of integrals

Filed under: Giải Tích 6 (MA5205), PDEs — Tags: — Ngô Quốc Anh @ 10:52

Long time ago, we studied [here] the following fact

Suppose $f \in L^1(\mathbb R^n) \cap L_{loc}^\infty (\mathbb R^n)$ with $f \geq 0$. Define

$\displaystyle Sf\left( x \right) = \int_{\mathbb{R}^n } {\log \frac{{\left| y \right|}}{{\left| {x - y} \right|}}f\left( y \right)dy}$.

Show that $Sf(x)$ is finite for all $x \in \mathbb R^n$ and $Sf \in L_{loc}^1(\mathbb R^n)$.

In this entry, from now on we continue to prove several useful results appearing in PDE. We shall prove the following

Theorem. Assume $u$ is a solution to

$\displaystyle (-\Delta)^\frac{3}{2} u(x)=-2e^{3u(x)}, \quad x \in \mathbb R^3$

with finite energy

$\displaystyle \int_{{\mathbb{R}^3}} {{e^{3u(x)}}dx} < \infty$.

Then

$\displaystyle\mathop {\lim }\limits_{|x| \to \infty } \frac{{u(x)}}{{\log |x|}} = - \frac{1}{{{\pi ^2}}}\int_{{\mathbb{R}^3}} {{e^{3u(y)}}dy}$.

## August 29, 2010

### Achieving regularity results via bootstrap argument, 4

Filed under: Giải Tích 6 (MA5205), PDEs — Tags: — Ngô Quốc Anh @ 2:36

Let us consider the following equation

$\displaystyle u(x) = \int_{{\mathbb{R}^n}} {\frac{{u{{(y)}^{\frac{{n + \alpha }}{{n - \alpha }}}}}}{{{{\left| {x - y} \right|}^{n - \alpha }}}}dy} , \quad x \in {\mathbb{R}^n}$

for $n\geqslant 1$ and $0<\alpha. In this entry, by using boothstrap argument, we show that

Theorem. If positive function $u \in L_{loc}^\frac{2n}{n-\alpha}(\mathbb R^n)$ solves the equation, then $u \in C^\infty(\mathbb R^n)$.

In the process of proving the result, we need the following result

Proposition. Let $V \in L^\frac{n}{\alpha}(B_3)$ be a non-negative function and set

$\displaystyle \delta(V)=\|V\|_{L^\frac{n}{\alpha}(B_3)}$.

For $\nu >r>\frac{n}{n-\alpha}$, there exist positive constants $\overline \delta<1$ and $C \geqslant 1$ depending only on $n, \alpha, r$ and $\nu$ such that for any $0 \leqslant V \in L^\frac{n}{\alpha}(B_3)$ with $\delta(V) \leqslant \overline \delta$, $h \in L^\nu(B_2)$ and $0 \leqslant u \in L^r(B_3)$ satisfying

$\displaystyle u(x) \leqslant \int_{{B_3}} {\frac{{V(y)u(y)}}{{{{\left| {x - y} \right|}^{n - \alpha }}}}dy} + h(x), \quad x \in {B_2}$

we have

$\displaystyle {\left\| u \right\|_{{L^\nu }({B_{1/2}})}} \leqslant C\left( {{{\left\| u \right\|}_{{L^r}({B_3})}} + {{\left\| h \right\|}_{{L^\nu }({B_2})}}} \right)$.

## July 24, 2010

### Regularity theory for integral equations

Filed under: Giải Tích 6 (MA5205), PDEs — Tags: — Ngô Quốc Anh @ 20:54

My purpose is to derive some regularity result concerning the following integral equation

$\displaystyle u(x) = \int_\Omega {\frac{{u(y)}}{{{{\left| {x - y} \right|}^{n - \alpha }}}}dy}$

where $\Omega \subset \mathbb R^n$ is open and bounded and $0<\alpha. To this purpose, in this entry we first consider the equation

$\displaystyle u(x) = \int_\Omega {\frac{{f(y)}}{{{{\left| {x - y} \right|}^{n - \alpha }}}}dy}$

for a suitable choice of $f$.

The case $f \in L^\infty(\Omega)$. We will prove that $u \in C^{1,\beta}(\Omega)$ for any $\beta\in (0,1)$. Indeed, up to a constant factor, the first derivative of $u$ are given by

$\displaystyle {D_i}u(x) = \int_\Omega {\frac{{{x_i} - {y_i}}}{{{{\left| {x - y} \right|}^{n + 2 - \alpha }}}}f(y)dy}$.

From this formula,

$\displaystyle\left| {{D_i}u({x^1}) - {D_i}u({x^2})} \right| = \mathop {\sup }\limits_\Omega |f|\int_\Omega {\left| {\frac{{x_i^1 - {y_i}}}{{{{\left| {{x^1} - y} \right|}^{n + 2 - \alpha }}}} - \frac{{x_i^2 - {y_i}}}{{{{\left| {{x^2} - y} \right|}^{n + 2 - \alpha }}}}} \right|dy}$.

By the intermediate value theorem, on the line from $x^1$ to $x^2$, there exists some $x^3$ with

$\displaystyle\left| {\frac{{x_i^1 - {y_i}}}{{{{\left| {{x^1} - y} \right|}^{n + 2 - \alpha }}}} - \frac{{x_i^2 - {y_i}}}{{{{\left| {{x^2} - y} \right|}^{n + 2 - \alpha }}}}} \right| \leqslant \frac{C}{{{{\left| {{x^3} - y} \right|}^{n + 2 - \alpha }}}}\left| {{x^1} - {x^2}} \right|$.

## June 9, 2010

### Invariance under fractional linear transformations

Filed under: Giải Tích 5, Giải Tích 6 (MA5205) — Tags: — Ngô Quốc Anh @ 5:02

I just read the following result due to Loss-Sloane published in J. Funct. Anal. this year [here]. This is just a lemma in their paper that I found very interesting.

Let $f$ be any function in $C_0^\infty(\mathbb R \setminus \{0\})$. Consider the inversion $x \mapsto \frac{1}{x}$ and set

$\displaystyle g(x) = {\left| x \right|^{\alpha - 1}}f\left( {\frac{1}{x}} \right)$.

Then $g \in C_0^\infty(\mathbb R)$ and

$\displaystyle\iint\limits_{{\mathbb{R}^2}} {\frac{{{{\left| {g(x) - g(y)} \right|}^2}}}{{{{\left| {x - y} \right|}^{\alpha + 1}}}}dxdy} = \iint\limits_{{\mathbb{R}^2}} {\frac{{{{\left| {f(x) - f(y)} \right|}^2}}}{{{{\left| {x - y} \right|}^{\alpha + 1}}}}dxdy}$.

Proof. For fixed $\varepsilon$ consider the regions

$\displaystyle {R_1}: = \left\{ {(x,y) \in {\mathbb{R}^2}:\left| {\frac{x}{y}} \right| > 1 + \varepsilon } \right\}$

and likewise,

$\displaystyle {R_2}: = \left\{ {(x,y) \in {\mathbb{R}^2}:\left| {\frac{y}{x}} \right| > 1 + \varepsilon } \right\}$.

## May 30, 2010

### Co-area formula for gradient, 2

Filed under: Giải Tích 6 (MA5205) — Tags: — Ngô Quốc Anh @ 20:29

Let us recall the following result

Theorem. Let $\Omega \subset \mathbb R^n$ be an open set and $u \in \mathcal D(\Omega)$. If $u \geqslant 0$ then for any $1 \leqslant p<\infty$, we have

$\displaystyle\int_\Omega|\nabla u|^p dx =\int_0^M \left(\int_{\{u=t\}}|\nabla u|^{p-1}d\sigma\right)dt$

where $M=\sup u$ over $\overline \Omega$.

The above result has been proven in this entry. If we chose $p=0$ then we would have

$\displaystyle\int_\Omega dx =\int_0^M \left(\int_{\{u=t\}}\frac{1}{|\nabla u|}d\sigma\right)dt$.

If we now replace $\Omega=\{u>t\}$, we get

$\displaystyle\int_{\{u>t\}}dx =\int_t^M \left(\int_{\{u=t\}}\frac{1}{|\nabla u|}d\sigma\right)dt$.

By differentiating with respect to $t$, we arrive at

$\displaystyle \frac{d}{dt}\int_{\{u>t\}}dx =-\int_{\{u=t\}}\frac{1}{|\nabla u|}d\sigma$.

In this entry, we shall prove the foregoing identity is actually true. This is the second interesting formula I have mentioned before. The proof, of course again, is based on a clever choice of test function.

## May 27, 2010

### A Simple Approach to the Hardy and Rellich inequalities

Filed under: Giải Tích 6 (MA5205) — Tags: , — Ngô Quốc Anh @ 16:07

The classical Hardy inequality in $\mathbb R^n$, $n \geqslant 3$, is stated as follows

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.

I suddenly found a very simple proof due to E. Mitidieri [here].

## May 24, 2010

Filed under: Giải Tích 6 (MA5205) — Tags: — Ngô Quốc Anh @ 18:55

I found two interesting formulas related to co-area formula while reading some tricks done by Talenti regarding to the best constant of the Sobolev inequality. The first result is to derive a representation of

$\displaystyle \int_\Omega |\nabla u|^pdx$

and the second result is to deal with differentiation of level sets. Having all these stuffs, I will derive a very short and beautiful proof concerning the lower bound of $\int \exp(u) dx$ where $u$, a positive solution to the following PDE

$\displaystyle-\Delta u = e^u$

This proof I firstly learned from a paper published in Duke Math. J. in 1991 by W. Cheng and C. Li [here].

Co-area formula. Suppose that $\Omega$ is an open set in $\mathbb R^n$, and $u$ is a real-valued Lipschitz function on $\Omega$. Then, for an integrable function $g$

$\displaystyle\int_\Omega g(x) |\nabla u(x)| dx = \int_{-\infty}^\infty \left(\int_{\{u=t\}}g(x) dH_{n-1}(x)\right)dt$

where $H^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure.

The Sard theorem. Let $f :\mathbb{R}^n \rightarrow \mathbb{R}^m$ be $C^k$, $k$ times continuously differentiable, where $k \geqslant \max\{n-m+1, 1\}$. Let $X$ be the critical set of $f$, the set of points $x$ in $\mathbb R^n$ at which the Jacobian matrix of $f$ has ${\rm rank} < m$. Then $f(X)$ has Lebesgue measure $0$ in $\mathbb R^m$.

The Sard theorem has some useful applications. For example, if $u \in \mathcal D(\Omega)$ the space of test functions where $\Omega \subset \mathbb R^n$, then for almost every $t$ in the range of $u$, we have that $|\nabla u|\ne 0$ on the level set $\{u=t\}$. Thus that level set will be an $(n-1)$-dimensional surface. Furthermore

$\{u=t\}=\partial \{u>t\}$

and

$|\{u=t\}|=0$.

Theorem. Let $\Omega \subset \mathbb R^n$ be an open set and $u \in \mathcal D(\Omega)$. If $u \geqslant 0$ then for any $1 \leqslant p<\infty$, we have

$\displaystyle\int_\Omega|\nabla u|^p dx =\int_0^M \left(\int_{\{u=t\}}|\nabla u|^{p-1}d\sigma\right)dt$

where $M=\sup u$ over $\overline \Omega$.

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