# Ngô Quốc Anh

### Talks

2016

1. Radial solutions of $\Delta^2 u + u^{-q} = 0$ in $\mathbb R^3$ with exactly quadratic growth at infinity [Mar 2016, VIASM, Vietnam].
2. Solutions to Toda systems on the plane via rational curves in $\mathbb CP^n$ [Jun 2016, VIASM, Vietnam].
3. Q-curvature flow on manifolds of even dimension [July 2016, SISSA, Italy].

2015

1. A flow approach to the prescribed scalar curvature problem with vanishing Yamabe invariant [Sep 2015, KU, Korea].

2014

1. Lichnerowicz equations on compact Riemannian manifolds with or without boundary [Sep 2014, IECL, France].
2. What makes the Yamabe problem so interesting? [Nov 2014, VIASM, Vietnam].
3. Einstein constraint equations on Riemannian manifolds [Dec 2014, IMS, Singapore]:
Starting from the Einstein equation in general relativity, we carefully derive the Einstein constraint equations which specify initial data for the Cauchy problem for the Einstein equation. Then we show how to use the conformal method to study these constraint equations.
Part I: Basics of spacetimes.
Part II: Constraints and evolutions.
Part III: Conformal method and transformed PDEs.
Part IV: Solving Lichnerowicz equations.

2013

1. The Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds [Mar 2013, LMPT, France]. While the conformal method can be effectively applied for solving the Einstein constraint equations in most cases, it should be pointed out that there are several cases for which either partial result or no result was achieved, especially when gravity is coupled to field sources. Based on a division recently obtained by Y. Choquet-Bruhat, J. Isenberg, and D. Pollack, one can observe that there are two cases corresponding to the non-positive Yamabe-scalar field conformal invariant with sign-changing nonlinearities, for which no result was achieved. This is basically due to the fact that the Einstein-scalar field Lichnerowicz equation includes terms which are not seen in other cases, which need more refined analysis. In this talk, we show how to overcome those difficulties for compact manifolds by developing a new approach that suits for our analysis.

2012

1. Nonlinear triharmonic equations with negative exponents [Oct  2012, VNU, Vietnam]. In this talk, we study global positive $C^6$ solutions of the geometrically interesting equation

$\displaystyle\Delta^3 u + u^{-3}=0$

in $\mathbb R^3$. In the first part of the talk, we show that any global positive $C^6$ solution of the equation which has cubic growth at infinity verifies the following integral equation

$\displaystyle u(x) =\frac{1}{{64\pi^2 }}\int_{\mathbb R^3} {|x - y|^3{u^{-3}}(y)dy} + p_2(x).$

where $p_2$ is a polynomial of order less than or equal to $2$. In the second part of the talk, by using the method of moving spheres, we show that up to constant multiple, translation, and dilation, all solutions of the above integral equation with the cubic growth are given by

$\displaystyle u(x) = {(1 + |x{|^2})^{\frac{3}{2}}}$

provided $\deg p_2 = 0$. As far as we know, this seems to be the first result for triharmonic equations with a negative exponent in the critical case.

2011-

1. Positive solutions for a class of semilinear elliptic systems via the dual variational method [2006, VNU, Vietnam]. In this talk, we consider the existence of non-trivial solutions for semilinear elliptic systems with $n$-equations on a bounded domain of $\mathbb{R}^N$, with zero Dirichlet boundary conditions

$-\Delta u + Au = f(u),$
$u\left| {_{\partial \Omega } } \right. = 0,$

where

$u = \left( {u_1 ,u_2 ,..,u_n } \right)$, $f\left( u \right) = \left( {f_1 \left( {u_1 } \right),f_2 \left( {u_2 } \right),..,f_n \left( {u_n } \right)} \right)$, ${f_k \left( {u_k } \right)}$ ($k=1,2,..,n$)

are nonlinear functions for $u_k$ defined in $\Omega$ and $A = \left( {a_{ij} } \right)_{n \times n}$ is a matrix of real entries satisfying $a_{ij} = a_{ji}$ for all $i \ne j$. Here, we use the dual variational method.

2. An application of dual variational method to semilinear elliptic systems on a bounded domain [Feb 2007, VNU, Vietnam].
3. Morse theory and several applications to partial differential equations [May 2007, VNU, Vietnam].
4. Scalar curvatures of manifolds with negative conformal invariant [Oct 2009,SNUS, Singapore].
5. The constraint equations for the Einstein-scalar field system on manifolds [Jan 2010, NUS, Singapore]. While much is understood about constant mean curvature solutions of the constraint equation for the vacuum Einstein, Einstein–Maxwell and Einstein-Yang-Mills fields, much less is known about the solutions of  constant mean curvature equation for the Einstein-scalar field. It is well known that the conformal method can be applied in the Einstein-scalar case. The difficulty is that  the Lichnerowicz equation includes terms which are not seen in other cases, and which need more refined analysis. In this talk, we show how to overcome some of these difficulties for a compact manifold. Our approach allows us to include the vacuum constraint equations with an arbitrary cosmological constant, the Yamabe equation and even (all cases of) the prescribed scalar curvature problem as special cases. We also discuss a conjecture on Einstein-scalar field constraints on asymptotically hyperboloidal case and other related questions.
6. The vacuum Einstein constraint equations with freely specified mean curvature [Oct  2010, NUS, Singapore]. The conformal method of solving the Einstein constraint equations is remarkably effective when the mean curvature is constant, and is remarkably recalcitrant when it is not. In this talk, we will discuss some progresses towards our understanding of the non-CMC case.
7. Existence results for the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds in the null case [Oct 14; Oct 21, 2011, NUS, Singapore]. While the conformal method can be effectively applied for solving the Einstein constraint equations in most cases, it should be pointed out that there are several cases for which either partial result or no result was achieved, especially when gravity is coupled to field sources. Based on a division recently obtained in <Class. Quantum Grav. 24 (2007), pp. 808-828>, one can observe that there are two cases corresponding to the non-positive Yamabe-scalar field conformal invariant with sign-changing nonlinearities, for which no result was achieved. This is basically due to the fact that the Einstein-scalar field Lichnerowicz equation includes terms which are not seen in other cases, which need more refined analysis. In this talk, we show how to overcome those difficulties for compact manifolds by developing a new approach that suits for our analysis. During the talk, some non-existence, existence, and multiplicity results are also presented.

Last update: July 23, 2016