Ngô Quốc Anh

April 20, 2016

Stereographic projection, 6

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 1:08

I want to propose an alternative way to calculate the Jacobian of the stereographic projection \mathcal S. In Cartesian coordinates  \xi=(\xi_1, \xi_2,...,\xi_{n+1}) on the sphere \mathbb S^n and x=(x_1,x_2,...,x_n) on the plane, the projection and its inverse are given by the formulas

\displaystyle\xi _i = \begin{cases} \dfrac{{2{x_i}}}{{1 + {{\left| x \right|}^2}}},&1 \leqslant i \leqslant n, \hfill \\ \dfrac{{{{\left| x \right|}^2} - 1}}{{1 + {{\left| x \right|}^2}}},&i = n + 1. \hfill \\ \end{cases}

and

\displaystyle {x_i} = \frac{{{\xi _i}}}{{1 - {\xi _{n + 1}}}}, \quad 1 \leqslant i \leqslant n.

It is well-known that the Jacobian of the stereographic projection \mathcal S: \xi \mapsto x is

\displaystyle \frac{\partial \xi}{\partial x} = {\left( {\frac{2}{{1 + {{\left| x \right|}^2}}}} \right)^n}.

The way to calculate its Jacobian is to compare the ratio of volumes. First pick two arbitrary points x, y \in \mathbb R^n and denote \xi = \mathcal S(x) and \eta = \mathcal S(y).

The Euclidean distance between \xi and \eta is

\displaystyle |\xi -\eta|^2 = \sum_{i=1}^{n+1} |\xi_i - \eta_i|^2 =\sum_{i=1}^n |\xi_i - \eta_i|^2+|\xi_{n+1} - \eta_{n+1}|^2.

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August 29, 2014

Prescribed Q-curvature and scalar curvature problems in the null case

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 20:24

On a 2-dimensional compact Riemannian manifold (M, g) without boundary, the prescribed scalar curvature problem in the flat case is equivalent to solving the following PDE

\displaystyle -\Delta_g u = fe^{2u}

with f is a given non-constant smooth function on M and \Delta_g is the Laplace-Beltrami operator associated with the metric g.

Simply by integrating both sides of the PDE, it is immediate to see that if u solves the PDE, it would satisfy \int_M f e^{2u} dv =0; hence the candidate function f must change sign in M. In their elegant paper published in 1974, Kazdan and Warner showed that in addition to the sign-changing property of f, it must also satisfy the following inequality

\displaystyle \int_M f dv <0.

This is just a simple observation from integration by parts if we multiply both sides of the PDE by e^{-2u}. Interestingly, Kazdan and Warner were able to show that the above two properties are also sufficient in the sense that it is enough to prove that the PDE is solvable.

In higher dimensions, perhaps, the most natural generalization of the operator \Delta_g is the well-known Paneitz operator and its corresponding Q-curvature, see this link.

Assume that (M,g) is a compact Riemannian 4-manifold without boundary. We denote by P_g^4 the so-called Paneitz operator acting on any smooth function u via the following rule

\displaystyle P_g^4(u) = \Delta _g^2u + {\rm div}\left( {\frac{2}{3}{R_g} - 2{\rm Ric}_g} \right)du ,

where by {\rm Ric} and R we mean the Ricci tensor and the scalar curvature of g, respectively.

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July 21, 2014

A strong maximum principle due to Tolksdorf and application to the prescribed scalar curvature equation

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 21:08

In this note, we consider a strong maximum principle due to Tolksdorf and application to the prescribed scalar curvature equation. In the context of closed and compact Riemannian manifolds (M,g) of dimension n, it can be stated as follows

Theorem (Maximum Principle). Let (M,g) be a compact Riemannian manifold of dimension n. Let u \in C^1(M) be such that

\displaystyle -\Delta_g u \geqslant f(\cdot, u)

on M where f is a function such that

\displaystyle \partial_u f \leqslant 0

(i.e. f is monotone increasing w.r.t the variable u) and that

\displaystyle |f(x,r)| \leqslant C(K+|r|)|r|

for all (x,r) \in M \times \mathbb R and for some constant C>0. If u \geqslant 0 in M and u does not vanish identically, then u>0 in M.

To prove this theorem, we need two auxiliary results. The first result is the weak comparison principle which can be stated as the following.

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April 16, 2014

Locally H^1-bounded implies pointwise upper bounded for the prescribed Gaussian curvature equations

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 15:58

I want to continue my previous post on the prescribed Gaussian curvature equations. Still borrowing the idea and technique introduced in the Struwe et al’ paper, today, I want to talk about how one can pass from locally H^1-bounded to pointwise bounded. As always, we are interested in solving the following PDE

\displaystyle -\Delta u + k = K_\lambda e^{2u}.

For the sake of clarity, let say K_\lambda \equiv K_i \searrow K as i \to \infty and suppose for each n, u_i solves the PDE, i.e. the following

\displaystyle -\Delta u_i + k = K_i e^{2u_i}

holds. As we have already seen, the sequence of solution \{u_i\}_i is H^1-bounded in the region M_-=\{x \in M: K(x) <0\}. We now show that such an H^1-boundedness can guarantee that \{u_i\}_i is pointwise bounded from above in M_-. As we shall see later, perhaps, the argument used below only works for the sequence of solutions of the PDE.

To see this, it suffices to prove that

\displaystyle u_i \leqslant C(B)

for any but fixed ball B \subset \overline B \subset M_-. To see this, we first observe from the Trudinger inequality that for each p>2

\displaystyle\int_B {\exp (pu)d{v_g}} \leqslant c\exp \left[ {\eta \frac{{{p^2}}}{4} \int_B |\nabla u|^2 dv_g} + \frac{1}{{\rm vol}(B)}\int_B u dv_g\right]

for some \eta,c>0. Note that

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April 15, 2014

Locally H^1-bounded for the Palais-Smale sequences in the region when the Gaussian curvature candidate is negative

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 16:22

In 1993, Ding and Liu announced their result about the multiplicity of solutions to the prescribed Gaussian curvature on compact Riemannian 2-manifolds of negative Euler characteristic. Their interesting result then published in the journal Trans. Amer. Math. Soc. in 1995, see this link.

Roughly speaking, by starting with the prescribed Gaussian curvature equation, i.e.

\displaystyle -\Delta u + k = Ke^{2u}

when the Euler characteristic \chi(M) is negative, i.e.

\displaystyle 2\pi \chi (M) = \int_M k e^{2u}dv_g <0,

they perturbed K using

\displaystyle K_\lambda = K+\lambda

where \lambda is a real number and the candidate function K is assumed to be

\displaystyle \max_{x \in M} K(x)=0.

Then they were interested in solving the following PDE

\displaystyle -\Delta u + k = K_\lambda e^{2u}.

Their main result can be stated as follows:

Theorem (Ding-Liu). There exists a \lambda^\star > 0 such that

  • the PDE has a unique solution for \lambda \leqslant 0;
  • the PDE has at least two solutions if 0<\lambda<\lambda^\star; and
  • the PDE has at least one solution when \lambda = \lambda^\star.

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March 26, 2014

Two pointwise conformal metrics having the same Ricci tensor must be homothetic

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 13:06

The aim of this note is to recall the following interesting result by X. Xu published  in Proc. AMS in 1992, here.

Theorem. Suppose (M,g) is a compact, oriented Riemannian manifold without boundary of dimension n \geq 2. If \widehat g=e^{2\varphi} g and \text{Ric}(\widehat g)=\text{Ric}(g), then \varphi is constant. In other words, two pointwise conformal metrics that have the same Ricci tensor must be homothetic.

A proof for this result is quite simple. First, we recall the following conformal change

\displaystyle\widehat{\text{Ric}}_{ij}=\text{Ric}_{ij} -(n-2)\big( \text{Hess}(\varphi)_{ij}-\nabla_i\varphi \nabla_j\varphi \big) - \big( \Delta_g \varphi + (n-2) |\nabla \varphi|^2 \big) g_{ij}

where \Delta_g u = \text{div}(\nabla u). Therefore, if \text{Ric}(\widehat g)=\text{Ric}(g), then we obtain the following fact

\displaystyle (n-2)\big( \text{Hess}(\varphi)_{ij}-\nabla_i\varphi \nabla_j\varphi \big)+ \big( \Delta_g \varphi + (n-2) |\nabla \varphi|^2 \big) g_{ij}=0.

However, the term \nabla_i\varphi \nabla_j\varphi appearing in the preceding identity seems to be bad. To avoid it, the author used the following conformal change

\displaystyle \widehat g = \frac{1}{u^2} g

for some positive function u, i.e. \varphi = -\log u or u=e^{-\varphi}. Then we calculate to obtain

\displaystyle \nabla_i u =-e^\varphi \nabla_i \varphi

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March 5, 2014

Uniformly upper Bound for Positive Smooth Solutions To The Lichnerowicz Equation In R^N

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 16:16

In this note, we are interested in the following Lichnerowicz type equation in \mathbb R^n

\displaystyle -\Delta u =-u^q+\frac{1}{u^{q+2}}, \quad q>0.

As we have already seen from the previous note that solutions for the above equation are always bounded from below for certain q.

Theorem (Brezis). Any solution of the Lichnerowicz equation with q>0 satisfies u\geqslant 1 in \mathbb R^n.

Remarkably, we are able to prove that solutions for the Lichnerowicz type equation are also bounded from above if we require q>1 instead of q>0. This result is basically due to L. Ma and X. Xu, see this paper.

Theorem (Ma-Xu). Any solution of the Lichnerowicz type equation with q>1 is uniformly bounded from above in \mathbb R^n.

Combining the two theorem above, we conclude that any solution of the Lichnerowicz equation, i.e. q=(n+2)/(n-2), is uniformly bounded in \mathbb R^n.

The idea of the proof for Ma-Xu’s theorem is as follows: Denote

\displaystyle f(u)=u^{-q-2} - u^q.

Fix x_0 \in \mathbb R^n but arbitrary, we then look for a positive radial super-solution v(x)=v(|x|)>0 of the Lichnerowicz type equation on the ball B_R(x_0) with positive infinity boundary condition for some R to be specified. This is equivalent to finding v in such a way that

\displaystyle\begin{array}{rcl}\displaystyle-\Delta v &\geqslant& f(v), \qquad\text{ in } B_R(x_0),\\ v&\equiv& +\infty, \qquad\text{ on }\partial B_R(x_0).\end{array}

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February 22, 2014

An application of the Brezis-Li-Shafrir estimate to the limiting case of the prescribing Gaussian curvature problem in the negative case

Filed under: PDEs, Riemannian geometry — Ngô Quốc Anh @ 14:56

On a Riemannian surface M, let consider the following PDE

\displaystyle -\Delta u +\alpha = R(x)e^u

naturally arising from the prescribed Gaussian curvature problem. A simple variable change, one can assume that \alpha is a negative constant, see this. It follows from a very well-known result due to Kazdan and Warner that it is necessary to have

\displaystyle \int_M R(x) dx<0.

In addition, Kazdan and Warner also showed that if \int_M R(x) dx<0 and R changes sign, then there exists a number \alpha_0 \in (-\infty, 0) such that the above PDE is solvable for all \alpha > \alpha_0 but not if \alpha<\alpha_0. In fact, the number \alpha_0 can be characterized as follows

\displaystyle \alpha_0 = \inf\{\alpha : \text{the PDE is solvable}\}.

This can be easily seen from the the following comprising property: If the PDE is solvable for some \alpha_1, it is also solvable for any \alpha_2 \geqslant \alpha_1.

However, Kazdan and Warner did not tell us what happens when \alpha=\alpha_0. In an attempt to see what really happens when \alpha=\alpha_0, Chen and Li made use of the Brezis-Li-Shafrir estimate to answer in the following way: The PDE is also solvable even when \alpha=\alpha_0. The purpose of this note is to talk about the beautiful Chen-Li argument, see this.

The idea is to approximate the equation for \alpha_0 by a sequence \{\alpha_k\}_k of negative real numbers in the following sense \alpha _k\searrow a_0 as k \to \infty. Their proof consists of three steps as follows:

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January 8, 2014

Conformal metric having strictly negative scalar curvature in a given region

Filed under: PDEs, Riemannian geometry — Ngô Quốc Anh @ 0:39

Let (M,g) be a closed (compact without boundary) Riemannian manifold of dimension 3 with a C^2 metric g of arbitrary Yamabe class. Suppose \Omega \subset M be an open subset of M with regular boundary \partial\Omega and with M\backslash (\Omega \cup \partial \Omega ) non-empty and open in M.NegativeScalarCurvatureInOmega

In this note, we mention a very interesting result basically due to O’Murchadha-York from here and Isenberg from here. The result says that there exists a conformal metric \widehat g \in [g] such that

\displaystyle \text{Scal}_{\widehat g} < -\xi<0

in \Omega for some constant \xi>0 to be specify later. The novelty of this result is that although the metric g may be of positive Yamabe class which tells us that it is impossible to construct a conformal metric which is everywhere negative, it is possible to make it negative in a proper subset of M. A proof for this result goes as follows:

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January 4, 2014

A Picone type identity for bi-Laplacian

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

Simultaneously, I have recently found the following identity in the same fashion of the Picone identity for \Delta. It says that

\displaystyle \left( \Delta u -\frac uv \Delta v\right)^2-\frac {2\Delta v}{v}\left| \nabla u - \frac uv \nabla v\right|^2 = |\Delta u|^2 -\Delta \left( \frac {u^2}v\right)\Delta v

for any function v \ne 0. It is worth noticing that the original Picone identity says that

\displaystyle \left| \nabla u - \frac{u}{v}\nabla v\right|^2= \left|\nabla u\right|^2 - \nabla \left( \frac{u^2}{v} \right) \cdot \nabla v \geqslant 0

for any function v \ne 0. It turns out that a few days ago, this identity appeared in a recent notes by Dwivedi  and Tyagi, see Lemma 2.1 from here. The extra term

\displaystyle\frac {2\Delta v}{v}\left| \nabla u - \frac uv \nabla v\right|^2

naturally appears since it only involves up to third order derivatives. However, to compare this term with 0, we only need to assume that \Delta v has a fixed sign. To see how this identity could be useful, let us consider the following equation

\displaystyle (-\Delta)^2 u +hu= f u ^\frac {n+4}{n-4}, \quad u>0, h<0

naturally arises from prescribing Q-curvature in Riemannian manifolds of dimension n \geqslant 5.

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