Ngô Quốc Anh

May 30, 2011

Norm of traceless Ricci tensor

Filed under: Riemannian geometry — Ngô Quốc Anh @ 15:06

In this note, we shall prove the following

\displaystyle {\left| {{\text{Ric}} - \dfrac{{\overline R }}{n}g} \right|^2} = |\mathop {{\text{Ric}}}\limits^ \circ {|^2} + \dfrac{1}{n}{(R - \overline R )^2}

that I have leart from this paper [here]. Here, R is scalar curvature, \mathop {{\text{Ric}}}\limits^ \circ is the traceless Ricci tensor and \overline R denotes the average of R. The proof is simple. First, we have

\displaystyle {\text{Ric}} - \frac{{\overline R }}{n}g = {\text{Ric}} - \frac{R}{n}g + \frac{{R - \overline R }}{n}g = \mathop {{\text{Ric}}}\limits^ \circ + \frac{{R - \overline R }}{n}g.

Therefore,

\displaystyle\begin{gathered} {\left| {{\text{Ric}} - \frac{{\overline R }}{n}g} \right|^2} = {g^{im}}{g^{jn}}{\left( {\mathop {{\text{Ric}}}\limits^ \circ + \frac{{R - \overline R }}{n}g} \right)_{ij}}{\left( {\mathop {{\text{Ric}}}\limits^ \circ + \frac{{R - \overline R }}{n}g} \right)_{mn}} \hfill \\ \qquad= {g^{im}}{g^{jn}}\left( {{{\mathop {{\text{Ric}}}\limits^ \circ }_{ij}}{{\mathop {{\text{Ric}}}\limits^ \circ }_{mn}} + \frac{1}{n}(R - \overline R )({g_{ij}}{{\mathop {{\text{Ric}}}\limits^ \circ }_{mn}} + {g_{mn}}{{\mathop {{\text{Ric}}}\limits^ \circ }_{ij}}) + \frac{1}{{{n^2}}}{{(R - \overline R )}^2}{g_{ij}}{g_{mn}}} \right). \hfill \\ \end{gathered}

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May 22, 2011

Conformal changes of Christoffel symbols

Filed under: Riemannian geometry — Ngô Quốc Anh @ 18:10

In this short note, we try to calculate conformal changes of Christoffel symbols that we have touched in the previous note [here]. Let us pick two Riemannian metrics g and \widetilde g on a manifold M sitting in the same conformal class, that is,

\displaystyle \widetilde g=e^{2f}g

where f is a smooth function on M.

Let us recall that Christoffel symbols are determined by

\displaystyle \Gamma _{ij}^k = \frac{1}{2}{g^{kl}}\left( {{g_{il,j}} + {g_{jl,i}} - {g_{ij,l}}} \right)

where

\displaystyle {g_{,m}} = \frac{{\partial g}}{{\partial {x^m}}}.

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May 19, 2011

The Ekeland variational principle

Filed under: PDEs — Ngô Quốc Anh @ 14:43

In mathematical analysis, Ekeland’s variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exists nearly optimal solutions to some optimization problems. Ekeland’s variational principle can be used when the lower level set of a minimization problems is not compact, so that the Bolzano–Weierstrass theorem can not be applied. Ekeland’s principle relies on the completeness of the metric space. Ekeland’s principle leads to a quick proof of the Caristi fixed point theorem.

Theorem (Ekeland’s variational principle). Let (X, d) be a complete metric space, and let F: X \to \mathbb R\cup \{+\infty\} be a lower semicontinuous functional on X that is bounded below and not identically equal to +\infty. Fix \varepsilon > 0 and a point u\in X such that

F(u) \leq \varepsilon + \inf_{x \in X} F(x).

Then there exists a point v\in X such that

  1. F(v) \leq F(u),
  2. d(u, v) \leq 1,
  3. and for all w \ne v, F(w) > F(v) - \varepsilon d(v, w).

Source: Wiki.

May 16, 2011

Some properties of the Yamabe equation in the null case

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 19:56

Let us consider the Yamabe equation in the null case, that is

\displaystyle -\Delta u = f u^\frac{n+2}{n-2}, \quad x \in M

where M is a compact manifold of dimension n without boundary. We assume that u>0 is a smooth positive solution.

Since the manifold is compact without the boundary, the most simple result is

\displaystyle\int_M f u^\frac{n+2}{n-2}=0

by integrating both sides of the equation. Now we prove that

\displaystyle\int_M f <0.

Indeed, multiplying both sides of the PDE with u^{-\frac{n+2}{n-2}} and integrating over M, one obtains

\displaystyle -\int_M(\Delta u) u^{-\frac{n+2}{n-2}} = \int_M f .

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May 14, 2011

The Nehari manifold

Filed under: PDEs — Ngô Quốc Anh @ 21:10

In the calculus of variations, a branch of mathematics, a Nehari manifold is a manifold of functions, whose definition is motivated by the work of Zeev Nehari (1960, 1961). It is a differential manifold associated to the Dirichlet problem for the semilinear elliptic partial differential equation

-\triangle u = |u|^{p-1}u,\text{ with }u\mid_{\partial(\Omega)} = 0.

Here \Delta is the Laplacian on a bounded domain \Omega in \mathbb R^n.

There are infinitely many solutions to this problem. Solutions are precisely the critical points for the energy functional

\displaystyle J(v) = \frac12\int_{\Omega}{\|\nabla v\|^2\,d\mu}-\frac1{p+1}\int_{\Omega}{|v|^{p+1}\,d\mu}

on the Sobolev space H^1_0(\Omega). The Nehari manifold is defined to be the set of v \in H^1_0(\Omega) such that

\displaystyle\|\nabla v\|^2_{L^2(\Omega)} = \|v\|^{p+1}_{L^{p+1}(\Omega)} > 0.

Solutions to the original variational problem that lie in the Nehari manifold are (constrained) minimizers of the energy, and so direct methods in the calculus of variations can be brought to bear.

More generally, given a suitable functional J, the associated Nehari manifold is defined as the set of functions u in an appropriate function space for which

\langle J'(u), u\rangle = 0.

Here J' is the functional derivative of J.

Source: Wiki.

May 11, 2011

On the uniqueness for the Yamabe problem

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 14:25

The Yamabe problem has been discussed here. Basically, starting from a metric g, for a given constant R Yamabe wanted to show there always exists a positive function \varphi such that the scalar curvature of metric \overline g defined to be \varphi^\frac{4}{n-2}g equals R. In terms of PDEs, the scalar curvature satisfies the equation (called Yamabe equation)

\displaystyle \frac{{4\left( {n - 1} \right)}} {{n - 2}}\Delta \varphi + R\varphi = \overline R {\varphi ^{\frac{{n + 2}} {{n - 2}}}}

where \overline R the scalar curvature of metric \overline g. It is not hard to see that in the negative and null cases, two solutions of the Yamabe equation with \overline R = {\rm const} are proportional. Let us prove the following

Theorem. In the negative and null cases, two solutions of the Yamabe equation with R=\overline R = {\rm const} are constant 1.

We first fix a background metric g_0 and let \varphi_0 be a solution to the Yamabe equation with \overline R=\mu. That is, the scalar curvature of metric \varphi_0^\frac{4}{n-2}g_0 equals \mu.

We now consider a new metric g_1 still sitting in the same conformal class [g_0] of g_0 with conformal factor \varphi_1 such that its scalar curvature equals \mu. In other words, \varphi_1 is also a solution of the Yamabe equation with R=\overline R=\mu. Keep in mind

\displaystyle \frac{{4\left( {n - 1} \right)}} {{n - 2}}\Delta \varphi_1 + \mu\varphi_1 = \mu_1 {\varphi_1 ^{\frac{{n + 2}} {{n - 2}}}}.

We have two cases.

  • Suppose \mu=0. From the equation above, \Delta \varphi_1=0, that is, \varphi_1 is constant.
  • Suppose \mu<0. At a point P where \varphi_1 is maximum, \Delta \varphi_1 \geqslant 0, thus \varphi_1(P) \leqslant 1. Similarly, at a point Q where \varphi_1 is minimum, \Delta \varphi_1 \leqslant 0, thus \varphi_1(Q) \geqslant 1. Consequently, \varphi_1 \equiv 1.

Notice that the uniqueness result in the positive  case is no longer true in general. Due to Obata, this is true for Einstein manifolds, that is, when the Ricci curvature and metric are proportional.

May 8, 2011

Commutator of Delta and Gradient on functions

Filed under: Riemannian geometry — Ngô Quốc Anh @ 0:05

Let us prove following interesting identity between \Delta and \nabla

\Delta \nabla_i f=\nabla_i \Delta f+{\rm Ric}_{ij}\nabla_j f

for any function f.

For any function f, using the formula

\Delta = g^{ij}\nabla_i \nabla_j

we obtain

\Delta {\nabla _i}f = {g^{mn}}{\nabla _m}{\nabla _n}({\nabla _i}f).

Since f is a funtion, we know that

{\nabla _n}({\nabla _i}f) = {\nabla _i}({\nabla _n}f)

then we obtain

\Delta {\nabla _i}f = {g^{mn}}{\nabla _m}({\nabla _i}({\nabla _n}f)).

Now \nabla_n f is a vector, we need to use the Riemmanian curvature tensor, which measures the difference between \nabla_m \nabla_i and \nabla_i\nabla_m.

{\nabla _m}({\nabla _i}({\nabla _n}f)) = {\nabla _i}({\nabla _m}({\nabla _n}f)) + {\rm Rm}({e^m},{e^i}){\nabla _n}f.

Notice that there was an extra term involving Lie bracket. Fortunately, that term vanishes. Thus

\Delta {\nabla _i}f = {g^{mn}}{\nabla _i}({\nabla _m}({\nabla _n}f)) + {g^{mn}}{\rm Rm}({e^m},{e^i}){\nabla _n}f = {\nabla _i}\Delta f + {\rm Ric}_{ik}{\nabla _k}f.

The proof follows.

May 6, 2011

Laplace-Beltrami vs. Laplacian

Filed under: Riemannian geometry — Ngô Quốc Anh @ 8:10

Let us do compare \Delta_g with respect to metric g and \Delta with respect to Euclidean metric. For simplicity, let us try with function u^{-\beta}. Obviously,

\displaystyle\begin{gathered} \Delta ({u^{ - \beta }}) = {\partial ^i}{\partial _i}({u^{ - \beta }}) \hfill \\ \qquad= {\partial ^i}( - \beta {u^{ - \beta - 1}}{\partial _i}u) \hfill \\ \qquad= \beta (\beta + 1){u^{ - \beta - 2}}({\partial ^i}u)({\partial _i}u) - \beta {u^{ - \beta - 1}}{\partial ^i}{\partial _i}u \hfill \\ \qquad= \beta (\beta + 1){u^{ - \beta - 2}}|\nabla u{|^2} - \beta {u^{ - \beta - 1}}\Delta u \hfill \\ \qquad= - \beta {u^{ - \beta - 1}}\left( {\Delta u - (\beta + 1)\frac{{|\nabla u{|^2}}}{u}} \right). \hfill \\ \end{gathered}

Next, by definition of \Delta_g we obtain

\displaystyle\begin{gathered} {\Delta _g}({u^{ - \beta }}) = \frac{1}{{\sqrt {\det g} }}{\partial _j}\left( {\sqrt {\det g} {g^{ij}}{\partial _i}({u^{ - \beta }})} \right) \hfill \\ \qquad= \frac{1}{{\sqrt {\det g} }}{\partial _j}\left( {( - \beta ){u^{ - \beta - 1}}\sqrt {\det g} {g^{ij}}{\partial _i}u} \right) \hfill \\ \qquad= \frac{1}{{\sqrt {\det g} }}\left[ {\beta (\beta + 1){u^{ - \beta - 2}}{\partial _ju}\sqrt {\det g} {g^{ij}}{\partial _i}u - \beta {u^{ - \beta - 1}}{\partial _j}\left( {\sqrt {\det g} {g^{ij}}{\partial _i}u} \right)} \right] \hfill \\ \qquad= \beta (\beta + 1){u^{ - \beta - 2}}|\nabla u{|_g^2} - \beta {u^{ - \beta - 1}}\Delta u_g \hfill \\ \qquad= - \beta {u^{ - \beta - 1}}\left( {\Delta_g u - (\beta + 1)\frac{{|\nabla u{|_g^2}}}{u}} \right). \hfill \\ \end{gathered}

Thus

\displaystyle {\Delta _g}({u^{ - \beta }}) = \Delta ({u^{ - \beta }}).

So the question is why do they equal?

May 3, 2011

A squeezing argument and application to logistic equations

Filed under: Uncategorized — Ngô Quốc Anh @ 1:19

The purpose of this note is to introduce the squeezing method. As an application, we prove some Liouville type theorem for solutions to logistic equations. Here the text is adapted from a paper by Du and Ma published in J. London Math. Soc. in 2001 [here].

Let prove the following theorem.

Theorem. Let \lambda \in \mathbb R and u \in C^2(\mathbb R^N) be a non-negative stationary solution of

u_t - \Delta u = \lambda u - u^p.

Then u must be a constant.

The basic ingredients in the proof consist of the following three lemmas. But first of all, let us denote by L a (uniformly) second order elliptic operator satisfying the PDE

-L(u)=\lambda a(x)u-b(x) u^p, \quad x\in \mathbb R^N

where

L(u)=\sum_{ij} (a_{ij}u_{x_i})_{x_j}

with a_{ij} smooth, a_{ij}=a_{ji} and

\sigma_1 |\xi|^2 \leqslant \sum_{ij}a_{ij}(x)\xi_i\xi_j \leqslant \sigma_2 |\xi|^2

for some positive constants \sigma_i, i=1,2.

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May 1, 2011

Method of moving planes: The cylindrically symmetric of solutions of critical Hardy–Sobolev operators

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 5:22

Recently, I have read a paper entitled “Classification of solutions of a critical Hardy-Sobolev operator” published in J. Differential Equations [here].

In that paper, the authors try to classify all positive solutions for the following equation

\displaystyle -\Delta u(x)=\frac{u(x)^\frac{n}{n-2}}{|y|} \quad \text{ in } \mathbb R^n

where x=(y,z)\in \mathbb R^k \times \mathbb R^{n-k}, and u \in \mathcal D^{1,2}(\mathbb R^n).

The main result of the paper can be formulated as follows

Theorem. Let u_0 be the function given by

\displaystyle u_0(x)=u_0(y,z)=c_{n,k}\left( (1+|y|)^2+|z|^2\right)^{-\frac{n-2}{2}}

where

\displaystyle c_{n,k}=\big((n-2)(k-1)\big)^\frac{n-2}{2}.

Then u is a solution to the equation above if and only if

\displaystyle u(y,z)=\lambda^\frac{n-2}{2}u_0(\lambda y, \lambda z+z_0)

for some \lambda>0 and some z_0 \in \mathbb R^{n-k}.

First, they use the method of moving planes to prove the cylindrically symmetric of solutions. Thanks to this symmetry, the equation reduces to an elliptic equation in the positive cone in R^2 which eventually leads to a complete identification of all the solutions of the equation. We skip the detailed discussion here and refer the reader to the original paper.

This result has recently been generalized by Cao and Li.

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