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.

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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.

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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).

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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)).

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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).

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