# Ngô Quốc Anh

## 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

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

## August 22, 2010

### Liouville’s theorem and related problems

Filed under: Giải tích 7 (MA4247), PDEs — Tags: — Ngô Quốc Anh @ 6:35

The following theorem is well-known

Theorem (Liouville). Let $\Omega$ be a simply connected domain in $\mathbb R^2$. Then all real solutions of

$\displaystyle \Delta u +2Ke^u=0$

in $\Omega$ where $K$ a constant, are of the form

$\displaystyle u=\log\frac{|f'|^2}{\left(1+\frac{K}{4}|f|^2\right)^2}$

where $f$ is a locally univalent meromorphic function in $\Omega$.

In geometry, our PDE

$\displaystyle \Delta u +2Ke^u=0$

says that under the case $\Omega=\mathbb R^2$, it holds

$e^u|dz|^2=f^*g_K$

where $g_K$ denotes the standard metric on $\mathbb S^2$ with constant curvature $K$. Thus we have

Corollary. All solutions of the PDE in $\mathbb R^2$ with $K>0$ and

$\displaystyle \int_{\mathbb R^2} e^u<\infty$

are of the form

$\displaystyle u(x)=\log\frac{16\lambda^2}{\left(4+\lambda^2K|x-x_0|^2\right)^2},\quad \lambda>0, \quad x_0 \in \mathbb R^2$.

## August 19, 2010

### L^infinity-boundedness for a single solution of -Delta u = Vexp(u)

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 7:00

The aim of this entry is to derive the $L^\infty$-boundedness for a single solution of the following PDE

$\displaystyle -\Delta u = V(x) e^u$

over a domain $\Omega$. This elegant result had been done by Brezis and Merle around 1991 published in Comm. Partial Differential Equations [here].

There are two possible cases.

The case of bounded domain. Let us assume $u$ a solution of the following PDE

$\displaystyle\begin{cases}- \Delta u = V(x){e^u}& \text{ in }\Omega , \hfill \\ u = 0&\text{ on }\partial \Omega ,\end{cases}$

where $\Omega \subset \mathbb R^2$ is a bounded domain and $V$ is a given function on $\Omega$.

Theorem. If $V \in L^p$ and $e^u \in L^{p'}$ for some $1 then $u \in L^\infty$.

Proof. It first follows from the Brezis-Meler inequality that

$e^{ku} \in L^1, \quad \forall k>0$

which by the Holder inequality gives

$e^{u} \in L^r, \quad \forall r<\infty$.

Therefore, if $p<\infty$

$Ve^u \in L^{p-\delta}, \quad \forall \delta>0$

while if $p=\infty$

$Ve^u \in L^r, \quad \forall r<\infty$.

Thus, a standard $L^p$-estimate argument from the elliptic theory implies that $u$ is bounded.

## August 17, 2010

### Evaluate complex integral via the Fourier transform

Filed under: Giải tích 7 (MA4247) — Tags: — Ngô Quốc Anh @ 5:56

As suggested from this topic, we are interested in evaluating the following complex integral

$\displaystyle G(t)=\mathop {\lim }\limits_{A \to \infty } \int\limits_{ - A}^A {{{\left( {\frac{{\sin x}} {x}} \right)}^2}{e^{itx}}dx}$.

The trick here is to use the Fourier transform. Thanks to ZY for teaching me this interesting technique.

In $\mathbb R$, the Fourier transform of function $f$, denoted by $\mathcal F[f]$, is defined to be

$\displaystyle \mathcal F[f](y) = \int_{ - \infty }^\infty {f(x){e^{ - 2\pi ixy}}dx}$.

If we apply the Fourier transform twice to a function, we get a spatially reversed version of the function. Precisely,

$\displaystyle\begin{gathered} \mathcal{F}\left[ {\mathcal{F}[f]} \right](z) = \int_{ - \infty }^\infty {\mathcal{F}[f](y){e^{ - 2\pi iyz}}dy} \hfill \\ \qquad\qquad= \int_{ - \infty }^\infty {\mathcal{F}[f](y){e^{2\pi iy( - z)}}dy} \hfill \\ \qquad\qquad= {\mathcal{F}^{ - 1}}\left[ {\mathcal{F}[f]} \right]( - z) \hfill \\ \qquad\qquad= f( - z) \hfill \\ \end{gathered}$

where $\mathcal F^{-1}$ denotes the inverse Fourier transform.

## August 16, 2010

### The Moser-Trudinger inequality for domains with holes

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 2:42

In this entry, we are interested in the following result

Theorem (Moser-Trudinger’s inequality for domains with holes). Let $\Omega$ be a bounded smooth domain in $\mathbb R^2$. Let $S_1$ and $S_2$ be two subsets of $\overline \Omega$ satisfying

${\rm dist}(S_1,S_2) \geqslant \delta_0>0$

and let $\gamma_0$ be a number satisfying $\gamma_0 \in \left(0,\frac{1}{2}\right)$. Then for any $\varepsilon>0$, there exists a constant $c=c(\varepsilon, \delta_0, \gamma_0)>0$ such that

$\displaystyle\int_\Omega {{e^u}} \leqslant C\exp \left[ {\frac{1}{{32\pi - \varepsilon }}\int_\Omega {{{\left| {\nabla u} \right|}^2}} + C} \right]$

holds for all $u \in H_0^1(\Omega)$ satisfying

$\displaystyle\frac{{\int_{{S_1}} {{e^u}} }}{{\int_\Omega {{e^u}} }} \geqslant {\gamma _0}, \quad \frac{{\int_{{S_2}} {{e^u}} }}{{\int_\Omega {{e^u}} }} \geqslant {\gamma _0}$.

## August 13, 2010

### An application of the (Moser-)Trudinger inequality to the mean field equations

Let $(M,g)$ be a compact Riemannian surface with the volume $|M|$. The simplest form of the mean field equation studied in the contexts of the prescribing Gaussian curvature, statistical mechanics of many vortex points in the perfect fluid and self-dual gauss theories is given by

$\displaystyle - {\Delta _g}u = \lambda \left( {\frac{{{e^u}}}{{\int_M {{e^u}d{v_g}} }} - \frac{1}{{|M|}}} \right), \quad \text{ on } M$

with

$\displaystyle\int_M {ud{v_g}} = 0$

where $\lambda$ is a real number.

The mean field equation has a variational structure, and $u$ is a solution if and only if it is a critical point of

$\displaystyle {J_\lambda }(v) = \frac{1}{2}\int_M {{{\left| {\nabla v} \right|}^2}d{v_g}} - \lambda \log \int_M {{e^v}d{v_g}}$

defined for $v \in H^1(M)$ with

$\displaystyle\int_M {vd{v_g}} = 0$.

It is worth noticing from this entry that so far our Moser-Trudinger’s inequality is just for $\mathbb S^2$

Theorem (Moser-Trudinger’s inequality for $\mathbb S^2$). There are constants $\eta>0$ and $c=c(g)>0$ such that for each $p \geqslant 2$

$\displaystyle \log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \left[ {\frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u } \right] + {C_2}$

for all $u \in W^{1,2}(\mathbb S^2)$.

## August 11, 2010

### Two ways to present SU(3) Toda systems

Filed under: PDEs, Riemannian geometry — Ngô Quốc Anh @ 4:13

This entry is concerned with SU(3) Toda systems in non-abelian relativistic self-dual gauge theory. Usually, they are given by

$\displaystyle\begin{gathered} - {\Delta _g}{u_1} = 2{\lambda _1}\left( {\frac{{{e^{{u_1}}}}}{{\int_M {{e^{{u_1}}}d{v_g}} }} - \frac{1}{{|M|}}} \right) - {\lambda _2}\left( {\frac{{{e^{{u_2}}}}}{{\int_M {{e^{{u_2}}}d{v_g}} }} - \frac{1}{{|M|}}} \right), \hfill \\ - {\Delta _g}{u_2} = - {\lambda _1}\left( {\frac{{{e^{{u_1}}}}}{{\int_M {{e^{{u_1}}}d{v_g}} }} - \frac{1}{{|M|}}} \right) + 2{\lambda _2}\left( {\frac{{{e^{{u_2}}}}}{{\int_M {{e^{{u_2}}}d{v_g}} }} - \frac{1}{{|M|}}} \right), \hfill \\ \end{gathered}$

on $M$ with

$\displaystyle\int_M {{u_1}d{v_g}} = \int_M {{u_2}d{v_g}} = 0$

where $(M,g)$ denotes a compact Riemannian surface and $\lambda_1$, $\lambda_2$ are non-negative constants.

We now consider function $h$ satisfying

$\displaystyle - \Delta h =\frac{1}{|M|} \quad \text{ in } M$

The existence of such a $h$ will be considered later.

We now introduce

$\displaystyle\begin{gathered} {v_1} = {u_1} - \log \int_M {{e^{{u_1}}}d{v_g}} - (2{\lambda _1} - {\lambda _2})h, \hfill \\ {v_2} = {u_2} - \log \int_M {{e^{{u_2}}}d{v_g}} - ( - {\lambda _1} + 2{\lambda _2})h. \hfill \\ \end{gathered}$

## August 8, 2010

### The nodal set of eigenfunctions

Filed under: PDEs — Ngô Quốc Anh @ 17:21

An eigenfunction $\varphi$ is meant to be a solution of Dirichlet’s problem

$\begin{cases}-\Delta \varphi = \lambda\varphi,&\text{ in } \Omega,\\\varphi=0,&\text{ on } \partial\Omega,\end{cases}$

where

$\displaystyle\Delta = \sum\limits_{k = 1}^n {\frac{{{\partial ^2}}}{{\partial x_k^2}}}$

is the Laplacian, $\Omega$ is a bounded smooth domain in $\mathbb R^n$, and $\lambda$ is a constant (i.e. the corresponding eigenvalue).

It is well known that the first eigenfunction is positive in $\Omega$, and all higher eigenfunctions must change sign.

Definition. The nodal set of an eigenfunction $\varphi$ is defined to be

$\displaystyle N = \overline {\left\{ {x \in \Omega :\varphi (x) = 0} \right\}}$.

To gain a better understanding of nodal sets let us consider a few examples.

Example 1. Let us consider the case when $\Omega=(0,l)$ where the first four eigenfunctions had been shown.

Red points are nodal sets. In this case, we simply call nodal nodes. This comes from the fact that all eigenvalues $\lambda_k$ and eigenfunctions $\varphi_k$ are already known

$\displaystyle{\varphi _k}(x) = \sqrt {\frac{2}{l}} \sin \left( {\frac{{n\pi }}{l}x} \right), \quad k \in \mathbb{N}$.

## August 6, 2010

### An upper bound for solutions via the maximum principle

Filed under: PDEs — Tags: , — Ngô Quốc Anh @ 1:29

It is known that [here] the following PDE

$\displaystyle\begin{cases}\Delta u =e^u, & {\rm in } \, \mathbb R^2,\\\displaystyle\int_{\mathbb R^2}e^u<\infty.\end{cases}$

has no $C^2$ solution. However, this is no longer true if we replace the whole space by a ball of radius $R$, say $B_R(0)$. In this entry, we show that if $u \in C^2(\overline B_R)$ is a solution of

$\Delta u=e^u, \quad {\rm in }\; B_R$

then

$\displaystyle u(0) \leqslant \log 8- 2\log R$.

To this purpose, let us recall the following

The Maximum Principle. Let assume $U\subset \mathbb R^2$ be open and bounded. We consider an elliptic operator $L$ of the form

$\displaystyle Lu = - \sum\limits_{i,j = 1}^2 {{a^{ij}}\frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}} + \sum\limits_{k = 1}^2 {{b^k}\frac{{\partial u}}{{\partial {x_k}}}} } + cu$

where coefficients are continuous and the standard uniform ellipticity condition holds.

## August 2, 2010

### The Brezis-Merle inequality

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 0:25

I am going to talk about uniform estimates and blow-up phenomena for solutions of

$\displaystyle -\Delta u=V(x)e^u$

in two dimensions done by Brezis and Merle around 1991 published in Comm. Partial Differential Equations [here]. As a first step, I am going to derive some inequality that we need later.

Assume $\Omega \subset \mathbb R^2$ is bounded domain and let $u$ be a solution of

$\displaystyle -\Delta u=f(x)$

together with Dirichlet boundary condition. Here function $f$ is assumed to be of class $L^1(\Omega)$.

Theorem (Brezis-Merle). For every $\delta \in (0,4\pi)$ we have

$\displaystyle\int_\Omega {\exp \left[ {\frac{{(4\pi - \delta )|u(x)|}}{{{{\left\| f \right\|}_1}}}} \right]dx} \leqslant \frac{{4{\pi ^2}}}{\delta }{\rm diam}{(\Omega )^2}$

where $\|\cdot\|_1$ denotes the $L^1$-norm and $u$ a solution to our PDE.