# Ngô Quốc Anh

## November 20, 2010

### The Harnack quantity and the local gradient estimate

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 1:27

We first recall the Bochner formula for a gradient vector field over a Riemannian manifold $(M,g)$ which is an important tool in geometric analysis

$\displaystyle\frac{1}{2}\Delta (|\nabla u{|^2}) = |{\nabla ^2}u{|^2} + \langle \nabla \Delta u,\nabla u\rangle + {\rm Ric}(\nabla u,\nabla u)$

which has already discussed [here] where $u$ a smooth function. Over $\mathbb R^n$ with standard Euclidean metric, the Bochner formula reads as follows

$\displaystyle\frac{1}{2}\Delta (|\nabla u{|^2}) = |{\nabla ^2}u{|^2} + \langle \nabla \Delta u,\nabla u\rangle$.

Assume $R_2>R>0$. Let $\phi$ be a cut-off function in $B_{R_2}(0)$ with $\phi=1$ on $B_R(0)$.

Definition. The following quantity

$P=\phi|\nabla w|^2$

is called the Harnack quantity where $w$ is some function.

## November 16, 2010

### The Kazdan-Warner identity

Filed under: Uncategorized — Ngô Quốc Anh @ 22:52

In this short note, we shall discuss a very beautiful identity named after Jerry L. Kazdan and F. W. Warner published in the Ann. of Math. (2) long time ago [here].

To be precise, let us consider the following partial differential equation

$\displaystyle \Delta u + h e^{2u}=c$

over a $2$-sphere $\mathbb S^2$ where $h$ is a function and $c$ is a constant. We shall prove the following

Theorem (Kazdan-Warner). It holds

$\displaystyle \int_{{\mathbb{S}^2}} {\left\langle {\nabla h,\nabla {x_j}} \right\rangle {e^{2u}}d{v_g}} = 2(1 - c)\int_{{\mathbb{S}^2}} {h{x_j}{e^{2u}}d{v_g}} , \quad j = 1,2,3,$

where $x_j$ are coordinates.

## November 9, 2010

### Characterization of biharmonic functions

Filed under: Uncategorized — Ngô Quốc Anh @ 16:30

It is well-known that if $B_R(x)$ is a ball with center $x$ and radius $R$ which is completely contained in the open set $\Omega\subset\mathbb R^n$, then the value $u(x)$ of a harmonic function (i.e. $\Delta u =0$) $u :\Omega\to\mathbb R$ at the center of the ball is given by the average value of $u$ on the surface of the ball; this average value is also equal to the average value of $u$ in the interior of the ball. In other words

$\displaystyle u(x) = \frac{1}{n\omega_n R^{n-1}}\int_{\partial B_R(x)} ud\sigma = \frac{1}{\omega_n R^n}\int_{B_R(x)} u(y)dy$

where $\omega_n$ is the volume of the unit ball in $n$ dimensions and $\sigma$ is the $n-1$ dimensional surface measure.

For a biharmonic function $u$ (i.e. $\Delta^2u=0$) we have a similar result due to Pizzetti.

Theorem. For any $n \in \mathbb N$, any solution $u$ of

$\Delta^2 u =0$ in $B_R(x) \subset \mathbb R^n$

satisfies

$\displaystyle u(x)-\frac{1}{\omega_n R^n}\int_{B_R(x)} u(y)dy=\frac{R^2}{2(n+2)}\Delta u(x)$.

## November 5, 2010

### What is a gradient flow?

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 18:40

In this entry, we shall discuss the following question: “What is a gradient flow?”.

Let $(H, \langle {\cdot, \cdot}\rangle)$ be a Hilbert space with norm $\| \cdot\|$. Let $\phi$ be a convex, lower semi-continuous functional defined on a dense domain $\text{Dom}(\phi) \subset H$.

Definition. The subdifferential of $\phi$ at a point $u \in \text{Dom}(\phi)$ is the set $\partial \phi(u) \subset H$ defined by

$\partial \phi (u) =\{p \in H: \phi(v) \geqslant \phi(u)+\langle p, v-u\rangle \quad \forall v \in \text{Dom}(\phi)\}$.

Vector $p$ is called subgradient at $u$, thus, the set of all subgradients at $u$ is called the subdifferential at $u$.

The geometric meaning of subdifferential is as the set of all possible “slopes” of affine hyperplanes touching the graph of $\phi$ from below at the point $u$. Thus, $\phi$ is differentiable at $u$ iff its subdifferential at $u$ contains exactly one vector as its derivative at that point.

We now recall the classical definition of gradient flow on a Hilbert space.

Definition. A function $u\in AC_{loc}((0,+\infty); H)$, the class of absolutely continuous from $(0,+\infty)$ to $H$, is a gradient flow of the convex, lower semi-continuous functional $\phi$ iff the differential inclusion

$u_t \in -\partial \phi(u(t))$

is satisfied almost everywhere with respect to $t$.

In practice for a given flow

$u_t = F(u)$,

if $\phi$ is differentiable and is given as an integral over some domain, say, $\Omega$, we simply verify there is some number $\lambda$ so that

$\displaystyle \int_\Omega u_t F(u) = \lambda \frac{d}{dt}\phi(u(t))$.

## November 2, 2010

### Jacobi’s formula for the differential of the determinant of matrices

Filed under: Các Bài Tập Nhỏ — Ngô Quốc Anh @ 15:34

In matrix calculus, Jacobi’s formula expresses the differential of the determinant of a matrix A in terms of the adjugate of A and the differential of A. The formula is

$\displaystyle d\, \mbox{det} (A) = \mbox{tr} (\mbox{adj}(A) \, dA)$.

It is named after the mathematician C.G.J. Jacobi.

We first prove a preliminary lemma.

Lemma. Given a pair of square matrices $A$ and $B$ of the same dimension $n$, then

$\displaystyle\sum_i \sum_j A_{ij} B_{ij} = \mbox{tr} (A^\top B)$.

Proof. The product $AB$ of the pair of matrices has components

$\displaystyle (AB)_{jk} = \sum_i A_{ji} B_{ik}$.

Replacing the matrix $A$ by its transpose $A^\top$ is equivalent to permuting the indices of its components

$\displaystyle (A^\top B)_{jk} = \sum_i A_{ij} B_{ik}$.

The result follows by taking the trace of both sides

$\displaystyle \mbox{tr} (A^\top B) = \sum_j (A^\top B)_{jj} = \sum_j \sum_i A_{ij} B_{ij} = \sum_i \sum_j A_{ij} B_{ij}$.

Theorem. It holds

$\displaystyle d \, \mbox{det} (A) = \mbox{tr} (\mbox{adj}(A) \, dA)$.