Ngô Quốc Anh

March 1, 2013

PhD Thesis: The Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds

Filed under: Luận Văn — Ngô Quốc Anh @ 6:12

Eventually, my PhD thesis had been released worldwide :).

Title: The Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds
Supervisor: XU XINGWANG
Keywords: Einstein-scalar field equation, Lichnerowicz equation, Critical exponent, Negative exponent, Conformal method, Variational method
Issue Date: 2012
Abstract: We establish some new existence and multiplicity results for positive solutions of the following Einstein-scalar field Lichnerowicz equations on compact manifolds (M,g) without the boundary of dimension n \geqslant 3,

\displaystyle -\Delta_g u + hu = fu^\frac{n+2}{n-2} + au^{-\frac{3n-2}{n-2}},

with either a negative, a zero, or a positive Yamabe-scalar field conformal invariant h. These equations arise from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity. The variational method can be naturally adopted to the analysis of the Hamiltonian constraint equations. However, it arises analytical difficulty, especially in the case when the prescribed scalar curvature-scalar field function f may change sign. To our knowledge, such a problem in its most generic case remains open. Finally, we establish some Liouville type results for a wider class of equations with constant coefficients including the Einstein-scalar field Lichnerowicz equation with constant coefficients.

Document Type: Thesis

If you are interested in my thesis, you can freely download it from the link

Thank you.

January 22, 2013

PhD Thesis Defense

Filed under: Linh Tinh, Luận Văn — Ngô Quốc Anh @ 15:46

I just passed my PhD defense on 18 Jan, 2013. Following is the front page of the slides I used during the defense.

The committee of my defense consists of

Since the thesis contains some unpublished results, I cannot provide the slides here but you can email me if interested.

Title of my PhD thesis defense

Title page of the slides used in my PhD thesis defense

April 11, 2012

Existence results for the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds

Filed under: Luận Văn, PDEs, Riemannian geometry — Tags: , — Ngô Quốc Anh @ 2:18

A couple of days ago, I got an acceptance for publication in Advances in Mathematics journal that makes me feel so exciting because of the prestige of the journal. This is part of my PhD thesis in NUS under the supervision of professor Xu. Besides, this is joint work with him.

The work looks like simple, I mean, we just try to solve the following PDE

\displaystyle {\Delta _g}u + hu = f{u^{{2^\star} - 1}} + \frac{a}{{{u^{{2^\star} +1}}}}, \quad u>0,

where \Delta_g=-{\rm div}_g(\nabla_g) is the Laplace-Beltrami operator, 2^\star=\frac{2n}{n-2} is the critical Sobolev exponent, M is a compact manifold without boundary of dimension n \geqslant 3, and h, f, a \geqslant 0 are smooth functions. In our work, the above PDE is numbered as (1.2). I don’t want to mention the physical background of the equation, in a few words, this equation is motivated by the Hamiltonian constraint equations of General Relativity through the so-called conformal method. Apparently, the important and frequently studied prescribing scalar curvature equation is just a particular case.

In this work, we focus on the negative Yamabe-scalar field invariant, that is, h<0. Our result basically consists of two theorems.

In the first result, we consider the case that f may change its sign, we prove

Theorem 1.1. Let (M,g) be a smooth compact Riemannian manifold without the boundary of dimension n \geqslant 3. Assume that f and a \geqslant 0 are smooth functions on M such that \int_M f dv_g<0\sup f > 0, \int_M a dv_g >0, and |h| < \lambda _f where \lambda_f is given in (2.1) below. Let us also suppose that the integral of a satisfies

\displaystyle\int_M {ad{v_g}} < \frac{1}{n-2}{\left( {\frac{{n - 1}}{n-2}} \right)^{n - 1}}{\left( {\frac{{|h|}}{{\int_M {|{f^ - }|d{v_g}} }}} \right)^n}\int_M {|{f^ - }|d{v_g}}

where f^- is the negative part of f. Then there exists a number C > 0 to be specified such that if

\displaystyle\frac{{\sup {f }}}{{\int_M {{|f^ -| }d{v_g}} }} <C,

Equation (1.2) possesses at least two smooth positive solutions.

In the next result, we consider the case that f \leqslant 0. In this case, we are able to get a complete characterization of the existence of solutions. More precisely, we prove

Theorem 1.2. Let (M,g) be a smooth compact Riemannian manifold without boundary of dimension n \geqslant 3. Let h<0 be a constant, f and a be smooth functions on M with a \geqslant 0 in M, f \leqslant 0 but not strictly negative. Then Equation (1.2) possesses one positive solution if and only if |h|<\lambda_f.

As one can see, the above theorem does not allow f to be strictly negative. Fortunately, our approach can cover this case too. This is the last remark in the paper as we prove the following: if f<0 then Equation (1.2) always possesses one positive solution, I mean, without any condition on f except the condition f<0.

It is important to note that in the case f \leqslant 0, the solution is always unique by using the monotone trick.

January 5, 2008

Luận Văn Thạc Sĩ

Filed under: Luận Văn — Ngô Quốc Anh @ 6:56

Tính giải được của một lớp hệ phương trình elliptic không tuyến tính




October 10, 2007

Luận Văn Cử Nhân

Filed under: Luận Văn — Ngô Quốc Anh @ 13:36

Bài toán Dirichlet đối với hệ elliptic nửa tuyến tính với phần chính là toán tử Laplace trong miền bị chặn




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