Ngô Quốc Anh

March 12, 2013

The generalized maximum principle

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 9:09

Let us start our notes with a very fundamental maximum principle for any strongly second order elliptic operator. We have

Theorem (Maximum principle). Let u satisfy the differential inequality

\displaystyle L[u] = \sum\limits_{i,j = 1}^n {{a_{ij}}(x)\frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}}} + \sum\limits_{k = 1}^n {{b_k}(x)\frac{{\partial u}}{{\partial {x_k}}}} \geqslant 0

in a domain D where L is uniformly elliptic. Suppose the coefficients a_{ij} and b_k are uniformly bounded. If u attains a maximum M at a point of D, then u = M in D.

In order to memorize the above result, let us think about the parabola y=x^2 with x\in[-1,1] and L=\Delta. In this one-dimentional case, L[y]=2 \geqslant 0 which confirms that y only achieves its maximum at x=\pm 1.

As you may know the operator L is only assumed to be strongly elliptic which only effects the coefficients a_{ij}. Regarding to the coefficients b_k, we only assume these are uniformly bounded. However, the uniform ellipticity of the operator L and the boundedness of the coefficients are not really essential as you can check in the proof. Besides, the domain D need not be bounded in this version.

Now for operators of the form (L + h), we still have a result analogous to the above.

Theorem (Maximum principle). Let u satisfy the differential inequality

\displaystyle (L + h)[u] >0

with h <0, with L uniformly elliptic in D, and with the coefficients of L and h bounded. If u attains a non-negative maximum M at an interior point of D, then u = M.

Clearly, the assumption h<0 and M \geqslant 0 are crucial as we may face some difficulty as raised in this note. Counterexamples are easily obtained if h > O. For example, the function u = \exp(-r^2) has an absolute maximum at r= 0.


March 9, 2013

On the integrability of the inverse of functions in the Sobolev space H_0^1

Filed under: PDEs — Ngô Quốc Anh @ 9:05

Yesterday, I read a recent paper by S. Yijing and Z. Duanzhi published in the journal Calculus of Variations in 2013. In one of their results, they proved the following

Theorem 2. Let \Omega \subset\mathbb R^N, N \geq 3 be bounded open set with smooth boundary. If p\geq 3, then

\displaystyle \int_\Omega |u|^{1-p}dx =+\infty, \quad \forall u \in H_0^1(\Omega).

Since p \geq 3, it is clear that 1-p \leq -2. By definition, \int_\Omega |u|^2dx <+\infty which gives us some comparison between u and its inverse power.

In order to prove the above theorem, the authors first proved the following

Theorem 1. Let \Omega \subset\mathbb R^N, N \geq 3 be bounded open set with smooth boundary, the function h\in L^1(\Omega) is positive a.e. in \Omega, and p>1. Then the equation

\displaystyle\begin{array}{rcl} \Delta u + h(x){u^{ - p}} &=& 0 \quad \text{ in } \Omega \hfill \\ u &>& 0 \quad \text{ in } \Omega \\ u &=& 0 \quad \text{ on }\partial \Omega \end{array}

admits a unique H_0^1-solution if and only if there exists u_0 \in H_0^1(\Omega) such that

\displaystyle \int_\Omega h(x) |u_0|^{1-p}dx <+\infty.

The proof of Theorem 1 is variational which makes use of the Nehari manifold and the fibering map. Basically, the authors tried to minimize the functional


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.

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