# Ngô Quốc Anh

## April 25, 2013

### The Cauchy formula for repeated integration

Filed under: Các Bài Tập Nhỏ, Giải Tích 2 — Ngô Quốc Anh @ 23:41

The Cauchy formula for repeated integration, named after Augustin Louis Cauchy, allows one to compress $n$ antidifferentiations of a function into a single integral.

Let $f$ be a continuous function on the real line. Then the $n$-th repeated integral of $f$ based at $a$,

$\displaystyle f^{(-n)}(x) = \int_a^x \int_a^{\sigma_1} \cdots \int_a^{\sigma_{n-1}} f(\sigma_{n}) \, \mathrm{d}\sigma_{n} \cdots \, \mathrm{d}\sigma_2 \, \mathrm{d}\sigma_1,$

is given by single integration

$\displaystyle f^{(-n)}(x) = \frac{1}{(n-1)!} \int_a^x\left(x-t\right)^{n-1} f(t)\,\mathrm{d}t.$

A proof is given by induction. Since $f$ is continuous, the base case follows from the Fundamental theorem of calculus

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x} f^{(-1)}(x) = \frac{\mathrm{d}}{\mathrm{d}x}\int_a^x f(t)\,\mathrm{d}t = f(x);$

where

$\displaystyle f^{(-1)}(a) = \int_a^a f(t)\,\mathrm{d}t = 0.$

## April 18, 2013

### A lower bound for solutions involving distance functions

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 21:14

In this note, we discuss an useful lemma and its beautiful proof given by Brezis and Cabré in a paper published in Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. in 1998. The full article can be freely downloaded from here. Before saying anything further, let us state the lemma.

Lemma 3.2. Suppose that $h \geq 0$ belongs to $L^\infty (\Omega)$. Let $v$ be the solution of

$\left\{\begin{array}{rcl} - \Delta v &=& h \quad \text{ in }\Omega , \hfill \\ v &=& 0 \quad \text{ on }\partial \Omega . \hfill \\ \end{array}\right.$

Then

$\displaystyle\frac{{v(x)}}{{\text{dist}(x,\partial \Omega )}} \geqslant c\int_\Omega {h\text{dist}(x,\partial \Omega )} ,\qquad\forall x \in \Omega,$

where $c>0$ is a constant depending only on $\Omega$.

This type of estimate frequently uses in the literature. We now show the proof of the lemma.

## April 1, 2013

### A proof of the uniqueness of solutions of the Lichnerowicz equations in the compact case

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 4:38

In this small note, we aim to derive some uniqueness property of solutions of the following PDE

$\displaystyle -a\Delta_g u +\text{Scal}_g u =|\sigma|_g^2 u ^{-2\kappa-3}-b\tau^2 u^{2\kappa+1}$

on a compact manifold $(M,g)$ where $\kappa=\frac{2}{n-2}$, $a=2\kappa+4$, and $b=\frac{n-1}{n}$.

We assume that $\phi_1$ and $\phi_2$ are solutions of the above PDE. Setting $\phi=\frac{\phi_2}{\phi_1}$. We wish to prove that $\phi=1$.

Let us consider the following trick basically due to David Maxwell, see this paper. Let $\widetilde g= \phi_1^{2\kappa}g$. Then the well-known formula for the Laplace-Beltrami operator $\Delta_g$, which is

$\displaystyle {\Delta _g}u = \frac{1}{{\sqrt {\det g} }}{\partial _i}(\sqrt {\det g} {g^{ij}}{\partial _j}u)$

helps us to write

$\displaystyle {\Delta _{\widetilde g}}u = \phi _1^{ - 2\kappa } \Big({\Delta _g}u + \frac{2}{{{\phi _1}}}{\nabla _g}{\phi _1}{\nabla _g}u \Big).$

Consequently,

$\displaystyle {\Delta _g}\phi = \phi _1^{2\kappa }{\Delta _{\widetilde g}}\phi - \frac{2}{{{\phi _1}}}{\nabla _g}{\phi _1}{\nabla _g}\phi .$

We now calculate $- a{\Delta _g}{\phi _2} + {R_g}{\phi _2}$ as follows.