# Ngô Quốc Anh

## September 9, 2016

### Benefits of “complete” and “compact” for analysis on Riemannian manifolds

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 10:08

When working on Riemannian manifolds, it is commonly assumed that the manifold is complete and compact. (The case of non-compactness is also of interest too.) In this entry, let us review the role of completeness and compactness in this setting.

How important the completeness is? Let us recall that for given a Riemannian manifold $(M,g)$, what we have is a nice structure as well as an appropriate analysis on any tangent space $T_pM$. For a $C^1$-curve $\gamma : [a,b] \to M$ on $M$, the length of $\gamma$ is

$\displaystyle L(\gamma) = \int_a^b \sqrt{g(\gamma (t)) \langle \partial_t \gamma \big|_t, \partial_t \gamma \big|_t\rangle} dt$

where $\partial_t \gamma\big|_t \in T_{\gamma (t)}M$ a tangent vector. (Note that by using curves, the tangent vector $\partial_t \gamma\big|_t$ is being understood as follows

$\displaystyle \partial_t \gamma\big|_t (f) = (f \circ \gamma)'(t)$

for any differentiable function $f$ at $\gamma(t)$.) Length of piecewise $C^1$ curves can be defined as the sum of the lengths of its pieces. From this a distance on $M$ whose topology coincides with the old one on $M$ is given as follows

$\displaystyle d_g(x,y) = \inf_\gamma L(\gamma)$

where the infimum is taken on all over the set of all piecewise $C^1$ curves connecting $x$ and $y$.

## April 20, 2016

### Stereographic projection, 6

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 1:08

I want to propose an alternative way to calculate the Jacobian of the stereographic projection $\mathcal S$. In Cartesian coordinates  $\xi=(\xi_1, \xi_2,...,\xi_{n+1})$ on the sphere $\mathbb S^n$ and $x=(x_1,x_2,...,x_n)$ on the plane, the projection and its inverse are given by the formulas

$\displaystyle\xi _i = \begin{cases} \dfrac{{2{x_i}}}{{1 + {{\left| x \right|}^2}}},&1 \leqslant i \leqslant n, \hfill \\ \dfrac{{{{\left| x \right|}^2} - 1}}{{1 + {{\left| x \right|}^2}}},&i = n + 1. \hfill \\ \end{cases}$

and

$\displaystyle {x_i} = \frac{{{\xi _i}}}{{1 - {\xi _{n + 1}}}}, \quad 1 \leqslant i \leqslant n$.

It is well-known that the Jacobian of the stereographic projection $\mathcal S: \xi \mapsto x$ is

$\displaystyle \frac{\partial \xi}{\partial x} = {\left( {\frac{2}{{1 + {{\left| x \right|}^2}}}} \right)^n}.$

The way to calculate its Jacobian is to compare the ratio of volumes. First pick two arbitrary points $x, y \in \mathbb R^n$ and denote $\xi = \mathcal S(x)$ and $\eta = \mathcal S(y)$.

The Euclidean distance between $\xi$ and $\eta$ is

$\displaystyle |\xi -\eta|^2 = \sum_{i=1}^{n+1} |\xi_i - \eta_i|^2 =\sum_{i=1}^n |\xi_i - \eta_i|^2+|\xi_{n+1} - \eta_{n+1}|^2.$

## August 9, 2015

### The third fundamental lemma in the method of moving spheres

Filed under: Uncategorized — Ngô Quốc Anh @ 4:03

In this post, we proved the following result (appeared in a paper by Y.Y. Li published in J. Eur. Math. Soc. (2004))

Lemma 1. For $n \geqslant 1$ and $\nu \in \mathbb R$, let $f$ be a function defined on $\mathbb R^n$ and valued in $[-\infty, +\infty]$ satisfying

$\displaystyle {\left( {\frac{\lambda }{{|y - x|}}} \right)^\nu }f\left( {x + {\lambda ^2}\frac{{y - x}}{{|y - x{|^2}}}} \right) \leqslant f(y), \quad \forall |y - x| > \lambda > 0.$

Then $f$ is constant or $\pm \infty$.

Later, we considered the equality case in this post and proved the following result:

Lemma 2. Let $n\geqslant 1$, $\nu \in \mathbb R$ and $f \in C^0(\mathbb R^n)$. Suppose that for every $x \in \mathbb R^n$ there exists $\lambda(x)>0$ such that

$\displaystyle {\left( {\frac{\lambda(x) }{{|y - x|}}} \right)^\nu }f\left( {x + {\lambda(x) ^2}\frac{{y - x}}{{|y - x{|^2}}}} \right) =f(y), \quad \forall |y - x| > 0.$

Then for some $a \geqslant 0$, $d>0$ and $\overline x \in \mathbb R^n$

$\displaystyle f(x) = \pm a{\left( {\frac{1}{{d + {{\left| {x - \overline x } \right|}^2}}}} \right)^{\frac{\nu }{2}}}.$

In this post, we consider the third result which can be stated as follows:

## April 28, 2015

### On the simplicity of the first eigenvalue of elliptic systems with locally integrable weight

Filed under: Uncategorized — Ngô Quốc Anh @ 0:52

Of interest in this note is the simplicity of the first eigenvalue of the following problem

$\begin{array}{rcl}-\text{div}(h_1 |\nabla u|^{p-2}\nabla u) &=& \lambda |u|^{\alpha-1} |v|^{\beta-1}v \quad \text{ in } \Omega\\-\text{div}(h_2 |\nabla v|^{q-2}\nabla v) &=& \lambda |u|^{\alpha-1} |v|^{\beta-1}u \quad \text{ in } \Omega\\u &=&0 \quad \text{ on } \partial\Omega\\v &=&0 \quad \text{ on } \partial\Omega\end{array}$

where $1 \leqslant h_1, h_2 \in L_{\rm loc}^1 (\Omega)$ and $\alpha, \beta>0$ satisfy

$\displaystyle \frac \alpha p + \frac \beta q = 1$

with $p,q >1$. A simple variational argument shows that $\lambda$ exists and can be characterized by

$\lambda = \inf_{\Lambda} J(u,v)$

where

$\displaystyle J(u,v)=\frac \alpha p \int_\Omega h_1 |\nabla u|^p dx + \frac \beta q \int_\Omega h_2 |\nabla v|^q dx$

and

$\Lambda = \{(u,v) \in W_0^{1,p} (\Omega) \times W_0^{1,q} (\Omega) : \Lambda (u,v) = 1\}$

with

$\displaystyle \Lambda (u,v)= \int_\Omega |u|^{\alpha-1}|v|^{\beta-1} uv dx.$

## April 10, 2015

### Existence of antiderivative of discontinuous functions

Filed under: Uncategorized — Ngô Quốc Anh @ 0:17

It is well-known that every continuous functions admits antiderivative. In this note, we show how to prove existence of antiderivative of some discontinuous functions.

A typical example if the following function

$f(x)=\begin{cases}\sin \frac 1x, & \text{ if } x \ne 0,\\ 0, & \text{ if }x=0.\end{cases}$

By taking to different sequences $x_k = 1/(2k\pi)$ and $y_k = 1/(\pi/2 + 2k\pi)$ we immediately see that $f$ is discontinuous at $x=0$. However, we will show that $f$ admits $F$ as its antiderivative.

To this end, we first consider the following function

$G(x)=\begin{cases}x^2\cos \frac 1x, & \text{ if } x \ne 0,\\ 0, & \text{ if }x=0.\end{cases}$

First we show that $G$ is differentiable. Clearly whenever $x \ne 0$, we obtain

$\displaystyle G'(x)=\sin \frac 1x + 2x \cos \frac 1x.$

## April 5, 2015

### The set of continuous points of Riemann integrable functions is dense

Filed under: Uncategorized — Ngô Quốc Anh @ 15:03

In this note, we prove that the set of continuous point of Riemann integrable functions $f$ on some interval $[a,b]$ is dense in $[a,b]$. Our proof start with the following simple observation.

Lemma: Assume that $P=\{t_0=a,...,t_n=b\}$ is a partition of $[a,b]$ such that

$\displaystyle U(f,P)-L(f,P)<\frac{b-a}m$

for some $m$; then there exists some index $i$ such that $M_i-m_i < \frac 1m$ where $M_i$ and $m_i$ are the supremum and infimum of $f$ over the subinterval $[t_{i-1},t_i]$.

We now prove this result.

Proof of Lemma: By contradiction, we would have $M_i-m_i \geqslant \frac 1m$ for all $i$; hence

$\displaystyle \frac{b-a}m = \sum_{i} \frac{t_i-t_{i-1}}{m}\leqslant \sum_{i} \big(M_i-m_i\big)(t_i-t_{i-1})=U(f,P)-L(f,P),$

We now state our main result:

Theorem. Let $f$ be Riemann integrable over $[a,b]$. Define

$\displaystyle \Gamma = \{ x\in [a,b] : f \text{ is continuous at } x\}$

Then $\Gamma$ is dense in $[a,b]$.

## March 13, 2015

### Comparing topologies of normed spaces: The equivalency of norms and the convergence of sequences

Filed under: Uncategorized — Ngô Quốc Anh @ 23:01

The aim of this note is to derive some connections between topologies of normed spaces in terms of the equivalency of norms and the convergence of sequences.

Topological space and its topology: First, we start with a topological space, call $X$. Its topology, say $\mathcal T$ is the collection of subsets of $X$ which satisfies certain conditions. In the literature, each member of the collection $\mathcal T$ is called an open set.

Regarding to topologies we have the following basic facts:

• Given two topologies $\mathcal T_1$ and $\mathcal T_2$ on $X$, we say that $\mathcal T_1$ is stronger (or finer or richer) than $\mathcal T_2$ if $\mathcal T_2 \subset \mathcal T_1$.
• Given a sequence $(x_n)_n$ in $X$, we say that $x_n$ converges to $x$ in topology $\mathcal T$ of $X$ if for any neighborhood $V$ of $x$, there exists some large number $N$ such that $x_n \in V$ for all $n \geqslant N$. (Here by the neighborhood $V$ of $x$ we mean that there exists an open set $O$ of $X$, i.e. $O$ is a member of the topology $\mathcal T$, such that $x \in O \subset V$.)

The key ingredient to compare topologies is to make use of the identity map. In the following part, we state a result which shall be used frequently in this note.

Topologies under the identity map: Given two topologies $\mathcal T_1$ and $\mathcal T_2$ on a topological space $X$, we are interested in comparing $\mathcal T_1$ and $\mathcal T_2$ in terms of the identity map $\rm id : (X, \mathcal T_1) \to (X, \mathcal T_2)$.

Lemma 1. The identity map $\rm id : (X, \mathcal T_1) \to (X, \mathcal T_2)$ is continuous if and only if $\mathcal T_1$ is stronger than $\mathcal T_2$.

Proof.

The proof is relatively easy. Indeed, if the map $\rm id$ is continuous, then the preimage of any $O_2 \in \mathcal T_2$ is also a member of $\mathcal T_1$ which immediately implies that $\mathcal T_1$ includes $\mathcal T_2$.

Having Lemma 1 in hand, we now try to compare topologies using norms.

## February 25, 2015

### Continuous functions on subsets can be extended to the whole space: The Kirzbraun-Pucci theorem

Filed under: Uncategorized — Ngô Quốc Anh @ 1:22

Let $f$ be a continuous function defined on a set $E \subset \mathbb R^N$ with values in $\mathbb R$ and with modulus of continuity

$\displaystyle \omega_f (s) := \sup_{|x-y|\leqslant s,x,y\in E} |f(x) - f(y)| \quad s>0.$

Obviously, the function $s \mapsto \omega_f(s)$ is nonnegative and nondecreasing in $[0,+\infty)$.

Our first assumption is that $\omega_f$ is bounded from above in $[0, \infty)$ by some increasing, affine function; that is to say there exists some $a,b \in \mathbb R^+$ such that

$\displaystyle \omega_f (s) \leqslant a s +b \quad \forall s \geqslant 0$.

Associated with $\omega_f$ having the above first assumption is the concave modulus of continuity of $f$, i.e. some smallest concave function $c_f$ lies above $\omega_f$. Such the function $c_f$ can be easily constructed using the following

$\displaystyle c_f (s) = \inf_\ell \{\ell(s) : \ell \text{ is affine and } \ell \geqslant \omega_f \text{ in } [0,+\infty)\}.$

As can be easily seen, once $\omega_f$ can be bounded from above by some affine function, the concave modulus of continuity of $f$ exists and is well-defined.

By definition and the monotonicity of $\omega_f$, we obtain

$\displaystyle |f(x)-f(y)| \leqslant \omega_f (|x-y|) \leqslant c_f (|x-y|).$

In this note, we prove the following extension theorem.

Theorem (Kirzbraun-Pucci). Let $f$ be a real-valued, uniformly continuous function on a set $E \subset \mathbb R^N$ with modulus of continuity $\omega_f$ satisfying the first assumption. There exists a continuous function $\widetilde f$ defined on $\mathbb R^N$ that coincides with $f$ on $E$. Moreover, $f$ and $\widetilde f$ have the same concave modulus of continuity $c_f$ and

$\displaystyle \sup_{\mathbb R^N} \widetilde f = \sup_E f, \quad \inf_{\mathbb R^N} \widetilde f = \inf_E f.$

## February 22, 2015

### The conditions (NN), (P), (NN+) and (P+) associated to the Paneitz operator for 3-manifolds

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 18:54

Of recent interest is the prescribed Q-curvature on closed Riemannian manifolds since it involves high-order differential operators.

In a previous post, I have talked about prescribed Q-curvature on 4-manifolds. Recall that for 4-manifolds, this question is equivalent to finding a conformal metric $\widetilde g =e^{2u}g$ for which the Q-curvature of $\widetilde g$ equals the prescribed function $\widetilde Q$? That is to solving

$\displaystyle P_gu+2Q_g=2\widetilde Q e^{4u},$

where for any $g$, the so-called Paneitz operator $P_g$ acts on a smooth function $u$ on $M$ via

$\displaystyle {P_g}(u) = \Delta _g^2u - {\rm div}\left( {\frac{2}{3}{R_g} - 2{\rm Ric}_g} \right)du$

which plays a similar role as the Laplace operator in dimension two and the Q-curvature of $\widetilde g$ is given as follows

$\displaystyle Q_g=-\frac{1}{12}(\Delta\text{Scal}_g -\text{Scal}_g^2 +3|{\rm Ric}_g|^2).$

Sometimes, if we denote by $\delta$ the negative divergence, i.e. $\delta = - {\rm div}$, we obtain the following formula

$\displaystyle {P_g}(u) = \Delta _g^2u + \delta \left( {\frac{2}{3}{R_g} - 2{\rm Ric}_g} \right)du.$

Generically, for $n$-manifolds, we obtain

$\displaystyle Q_g=-\frac{1}{2(n-1)} \Big(\Delta\text{Scal}_g - \frac{n^3-4n^2+16n-16}{4(n-1)(n-2)^2} \text{Scal}_g^2+\frac{4(n-1)}{(n-2)^2} |{\rm Ric}_g|^2 \Big)$

and

$\displaystyle {P_g}(u) = \Delta _g^2u + {\rm div}\left( { a_n {R_g} + b_n {\rm Ric}_g} \right)du + \frac{n-4}{2} Q_g u,$

where $a_n = -((n-2)^2+4)/2(n-1)(n-2)$ and $b_n =4/(n-2)$.

## January 24, 2015

### Reversed Gronwall-Bellman’s inequality

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 23:01

In mathematics, Gronwall’s inequality (also called Grönwall’s lemma, Gronwall’s lemma or Gronwall–Bellman inequality) allows one to bound a function that is known to satisfy a certain differential or integral inequality by the solution of the corresponding differential or integral equation. The differential form was proven by Grönwall in 1919. The integral form was proven by Richard Bellman in 1943. A nonlinear generalization of the Gronwall–Bellman inequality is known as Bihari’s inequality.

First, we consider the Gronwall inequality.

Type 1. Bounds by integrals based on lower bound $a$.

Let $\beta$ and $u$ be real-valued continuous functions defined on $[a,b]$. If $u$ is differentiable in $(a,b)$ and satisfies the differential inequality

$\displaystyle u'(t) \leqslant \beta(t) u(t),$

then $u$ is bounded by the solution of the corresponding differential equation $y'(t) = \beta (t)y(t)$, that is to say

$\displaystyle \boxed{u(t) \leqslant u(a) \exp\biggl(\int_a^t \beta(s) ds\biggr)}$

for all $t \in [a,b]$.

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