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.

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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.

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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),

which gives us a contradiction.

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].

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