# Ngô Quốc Anh

## 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

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

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 $[a,b]$.