Ngô Quốc Anh

February 22, 2015

The conditions (NN), (P), (NN+) and (P+) associated to the Paneitz operator for 3-manifolds

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 18:54

Of recent interest is the prescribed Q-curvature on closed Riemannian manifolds since it involves high-order differential operators.

In a previous post, I have talked about prescribed Q-curvature on 4-manifolds. Recall that for 4-manifolds, this question is equivalent to finding a conformal metric \widetilde g =e^{2u}g for which the Q-curvature of \widetilde g equals the prescribed function \widetilde Q? That is to solving

\displaystyle P_gu+2Q_g=2\widetilde Q e^{4u},

where for any g, the so-called Paneitz operator P_g acts on a smooth function u on M via

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

which plays a similar role as the Laplace operator in dimension two and the Q-curvature of \widetilde g is given as follows

\displaystyle Q_g=-\frac{1}{12}(\Delta\text{Scal}_g -\text{Scal}_g^2 +3|{\rm Ric}_g|^2).

Sometimes, if we denote by \delta the negative divergence, i.e. \delta = - {\rm div}, we obtain the following formula

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

Generically, for n-manifolds, we obtain

\displaystyle Q_g=-\frac{1}{2(n-1)} \Big(\Delta\text{Scal}_g - \frac{n^3-4n^2+16n-16}{4(n-1)(n-2)^2} \text{Scal}_g^2+\frac{4(n-1)}{(n-2)^2} |{\rm Ric}_g|^2 \Big)

and

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

where a_n = -((n-2)^2+4)/2(n-1)(n-2) and b_n =4/(n-2).

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January 24, 2015

Reversed Gronwall-Bellman’s inequality

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 23:01

In mathematics, Gronwall’s inequality (also called Grönwall’s lemma, Gronwall’s lemma or Gronwall–Bellman inequality) allows one to bound a function that is known to satisfy a certain differential or integral inequality by the solution of the corresponding differential or integral equation. The differential form was proven by Grönwall in 1919. The integral form was proven by Richard Bellman in 1943. A nonlinear generalization of the Gronwall–Bellman inequality is known as Bihari’s inequality.

First, we consider the Gronwall inequality.

Type 1. Bounds by integrals based on lower bound a.

Let \beta and u be real-valued continuous functions defined on [a,b]. If u is differentiable in (a,b) and satisfies the differential inequality

\displaystyle u'(t) \leqslant \beta(t) u(t),

then u is bounded by the solution of the corresponding differential equation y'(t) = \beta (t)y(t), that is to say

\displaystyle \boxed{u(t) \leqslant u(a) \exp\biggl(\int_a^t \beta(s) ds\biggr)}

for all t \in [a,b].

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January 7, 2015

The failure of compact Rellich-Kondrachov embedding: Unbounded domains and critical exponents

Filed under: Uncategorized — Ngô Quốc Anh @ 19:49

In a very old entry, I talked about an extension of Rellich-Kondrachov theorem for embeddings between Sobolev spaces. For the sake of convenience, here is the statement of this extension:

Theorem (Extension of Rellich-Kondrachov for bounded domains). Let \Omega \subset \mathbb R^n be an open, bounded Lipschitz domain, and let  1 \leqslant p \leqslant mn. Set

\displaystyle p^\star := \frac{np}{n - mp}.

Then we have

\displaystyle W^{j+m, p} (\Omega) \hookrightarrow W^{j, q} (\Omega) for  1 \leqslant q \leqslant p^\star

and

\displaystyle W^{j+m, p} (\Omega) \hookrightarrow \hookrightarrow W^{j,q} (\Omega) for  1 \leqslant q < p^\star.

Clearly, when q=p^\star=\frac{np}{n - mp}, the above embedding is not compact, in general. In this context, we call the failure of compact Rellich-Kondrachov embedding due to critical exponents.

There is an other example of the failure of compact Rellich-Kondrachov embedding which is basically due to the unbounded domains. In this entry, we address counter-examples for these two lacks of compactness.

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December 31, 2014

Conformal change of the Laplace-Beltrami operator

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 23:55

Happy New Year 2015!

In the last entry in 2014, I talk about conformal change of the Laplace-Beltrami operator. Given (M,g) a Riemannian manifold of dimension n \geqslant 2. We denote \widetilde g = e^{2\varphi} g a conformal metric of g where the function \varphi is smooth.

Recall the following formula for the Laplace-Beltrami operator \Delta_g calculated with respect to the metric g:

\displaystyle \Delta_g = \frac{1}{\sqrt{|\det g|}} \frac{\partial}{\partial x_j} \Big( \sqrt{|\det g|} g^{ij} \frac{\partial}{\partial x^i} \Big).

where \det g is the determinant of g. Then, it is natural to consider the relation between \Delta_g and \Delta_{\widetilde g} in terms of \varphi. Recall that by \widetilde g = e^{2\varphi} g we mean, in local coordinates, the following

\displaystyle \widetilde g_{ij} = e^{2\varphi} g_{ij},

hence by taking the inverse, we obtain

\displaystyle \widetilde g^{ij} = e^{-2\varphi} g^{ij}.

Clearly,

\displaystyle\det {\widetilde g} = e^{2n \varphi}\det g,

hence

\displaystyle\sqrt{| \det {\widetilde g} |} = e^{n \varphi} \sqrt{ |\det g| }.

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December 21, 2014

Conformal Changes of the Green function for the conformal Laplacian

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 11:00

Long time ago, I talked about conformal changes for various geometric quantities on a given Riemannian manifold (M,g) of dimension n, see this post.

Frequently used in conformal geometry in general, or when solving the prescribed scalar curvature equation in particular, is the conformal Laplacian, defined as follows

\displaystyle L_g(u) = - \frac{n-1}{4(n-2)}\Delta_g u + \text{Scal}_g u

where \text{Scal}_g is the scalar curvature of the metric g. The operator L_g is conformal in the sense that any change of metric \widehat g = \varphi ^\frac{4}{n-2}g would give the following magic identity

\displaystyle L_{\widehat g} (u) =\varphi^{-\frac{n+2}{n-2}} L_g (\varphi u).

Associated to the conformal Laplacian operator L_g is the Green function, if exists, \mathbb G_{L,g}. Mathematically, the Green function \mathbb G_{L,g} is defined to be a continuous function

\mathbb G_{L,g} : M \times M \backslash \{(x,x) : x \in M\} \to \mathbb R

such that for any x\in M, \mathbb G_{L,g} (x, \cdot) \in L^1(M) and for any u \in C^2(M) and any x \in M, we have the following representation

\displaystyle u(x) = \int_M \mathbb G_{L,g}(x,y) L_g(u)(y) dv_g (y).

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December 4, 2014

Equations satisfied by standard bubbles and their derivatives in the Euclidean space

Filed under: Uncategorized — Ngô Quốc Anh @ 21:37

This note is purely involved calculation. In \mathbb R^n, let denote by V_{(x,\varepsilon)} (y) the standard bubbles given by

\displaystyle V_{(x,\varepsilon)} (y)= \left( \frac{\varepsilon}{\varepsilon^2+|y-x|^2}\right)^\frac{n-2}{2}.

I am trying to derive some PDE for which the bubbles V_{(x,\varepsilon)} solves.

1. First, we try to calculate \Delta V_{(x,\varepsilon)}. Clearly,

\begin{array}{lcl} {\partial _{{y_i}}}{V_{(x,\varepsilon )}}(y) &=& \displaystyle \frac{{n - 2}}{2}{\left( {\frac{\varepsilon }{{{\varepsilon ^2} + |y - x{|^2}}}} \right)^{ - 1}}{V_{(x,\varepsilon )}}(y){\partial _{{y_i}}}\left( {\frac{\varepsilon }{{{\varepsilon ^2} + |y - x{|^2}}}} \right) \hfill \\ &=& \displaystyle -\frac{{n - 2}}{2}{\left( {\frac{\varepsilon }{{{\varepsilon ^2} + |y - x{|^2}}}} \right)^{ - 1}}{V_{(x,\varepsilon )}}(y)\frac{{2\varepsilon ({y_i} - {x_i})}}{{{{({\varepsilon ^2} + |y - x{|^2})}^2}}} \hfill \\ &=& \displaystyle -(n - 2)\frac{{{y_i} - {x_i}}}{{{\varepsilon ^2} + |y - x{|^2}}}{V_{(x,\varepsilon )}}(y).\end{array}

Taking derivative again gives

\begin{array}{lcl} \partial _{{y_i}{y_i}}^2{V_{(x,\varepsilon )}}(y) &=& \displaystyle -(n - 2) {\partial _{{y_i}}}\left( {\frac{{{y_i} - {x_i}}}{{{\varepsilon ^2} + |y - x{|^2}}}{V_{(x,\varepsilon )}}(y)} \right) \hfill \\ &=& \displaystyle -(n - 2) {V_{(x,\varepsilon )}}(y){\partial _{{y_i}}}\left( {\frac{{{y_i} - {x_i}}}{{{\varepsilon ^2} + |y - x{|^2}}}} \right) - (n - 2)\frac{{{y_i} - {x_i}}}{{{\varepsilon ^2} + |y - x{|^2}}}{\partial _{{y_i}}}{V_{(x,\varepsilon )}}(y) \hfill \\ &=& \displaystyle -(n - 2) {I_1} - (n - 2) {I_2}.\end{array}

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November 4, 2014

Baire properties for open subspaces

Filed under: Giải Tích 1 — Tags: — Ngô Quốc Anh @ 7:53

This post deals with a classical problem in functional analysis: The Baire space. I am not going to reproduce what we can learn and read from wikipedia; however, to make the post self-contained, following is what the Baire space is.

Loosely speaking, a Baire space X is a topological space in which any one of the following three equivalent conditions is satisfied:

  1. Whenever the union of countably many closed subsets of X has an interior point, then one of the closed subsets must have an interior point, i.e. if

    \displaystyle \text{int}\Big(\bigcup_{n \geqslant 1} C_n \Big) \ne \emptyset,

    then \text{int}(C_n) \ne \emptyset for some n. Here by C we mean a closed subset in X.

  2. The union of every countable collection of closed sets with empty interior has empty interior, that is to say, i.e if \text{int}(C_n) = \emptyset for all n, then

    \displaystyle \text{int}\Big(\bigcup_{n \geqslant 1} C_n \Big) =\emptyset.

  3. Every intersection of countably many dense open sets is dense, i.e.

    \displaystyle \overline{\bigcap_{n \geqslant 1} O_n} = X

    provided \overline{O_n}= X for every n. Here by O we mean an open subset in X.

What I am going to do is to show that every open subset of a Baire space is itself a Baire space, of course, under the subspace topology inherited from X. Hence, at the very beginning, we assume throughout this topic that X is a Baire space; hence admits all three equivalent conditions above.

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October 12, 2014

Construction of non-radial solutions for a Lichnerowicz type equation in the whole space

Filed under: Uncategorized — Ngô Quốc Anh @ 1:23

Given q \in (0,1) and N \geqslant 2, in this note, we are interested in construction of non-radial solutions for the following Lichnerowicz type equation

\displaystyle -\Delta u = -u^q + u^{-q-2}

in the whole space \mathbb R^N.

In the previous post, we showed how to construct non-radial solutions of the following equation

\displaystyle -\Delta u = -u^q.

Clearly, this equation comes from the Lichnerowicz type equation by writing off the term with a negative exponent.

To start our construction and for simplicity, let us denote by f the following

\displaystyle f(t) = t^q - t^{-q-2},

then a simple calculation shows f'(t)=qt^{q-1} + (q+2)t^{-q-3} and f''(t)=q(q-1)t^{q-2} - (q+2)(q+3)t^{-q-4}. Hence, the function f is monotone increasing in [0,+\infty). Moreover, there exists a real number a>0 sufficiently large such that f>0 and f is concave in [a,+\infty). In addition, we can choose the number a even large in such a way that

\displaystyle \frac 1C f''(t) \leqslant f''(2t) \leqslant C f''(t)

for some constant C>0.

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September 11, 2014

Construction of non-radial solutions for the equation Δu=u^q with 0<q<1 in the whole space

Filed under: Uncategorized — Ngô Quốc Anh @ 3:01

Given q \in (0,1) and N \geqslant 2, in this note, we are interested in construction of non-radial solutions for the following equation

\displaystyle \Delta u = u^q

in the whole space \mathbb R^N. The construction is basically due to Louis Dupaigne and mainly depends on the unique radial solution of the equation.

To start our construction, let us recall that there is a unique radial solution, denoted by u_0, of the equation \Delta u = u^q such that u_0 (0)=1 and u'_0(0)=0. Moreover, u_0 is globally defined and blows up at infinity at a fixed rate

\displaystyle \lim_{r \to +\infty} \frac{u_0(r)}{r^\alpha} = L

where \alpha = \frac{2}{1-q} and L=[\alpha (\alpha + N-2)]^{-1/(q-1)}>0, see a paper by Yang and Guo published in J. Partial Diff. Eqns. in 2005.

Notice that

\displaystyle \Delta u_0 = r^{1-N} (r^{N-1} u'_0)'.

Hence, integrating both sides of the equation for u_0 gives

\displaystyle\frac{du_0}{dr} = r^{1-N} \int_0^r t^{N-1} u_0^q dt

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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.

(more…)

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