Ngô Quốc Anh

February 26, 2011

Decay estimates of a linear problem in R^2

Filed under: PDEs — Ngô Quốc Anh @ 2:36

I found the following simple result in a paper due to F. Lin and J.C. Wei published in Commun. Pure Applied Math. this year [here].

Lemma 4.2. Let h satisfy

-\Delta h=f(z), \quad h(\overline z)=-h(z), \quad |h| \leqslant C

where f satisfies

\displaystyle |f(z)| \leqslant \frac{C}{(1+|z|)^{2+\sigma}},\quad 0<\sigma<1.

Then

\displaystyle |h(z)| \leqslant \frac{C}{(1+|z|)^{\sigma}}.

Proof.

By the Poisson formula,

\displaystyle h(z)=\frac{1}{2\pi}\int_{\{y_2 >0\}}\log \frac{|\overline z - y|}{|z-y|}f(y)dy.

Because of the grown of f, it is known that h(z) \to 0 as |z| \to +\infty. We construct suitable supersolutions on \{x_2 > 0\}. Then the result will follow from the maximum principle.

In fact, let

h_0(z)=r^\beta x_2^\gamma

where r=|z| and the parameters are chosen so that

\beta+\gamma=-\sigma, \quad 0<\sigma<\gamma<1.

Then simple computations show that

\displaystyle\begin{gathered} \Delta {h_0} = \Delta ({r^\beta }x_2^\gamma ) \hfill \\ \qquad= {r^\beta }x_2^\gamma \left[ {({\beta ^2} + 2\beta \gamma ){r^{ - 2}} + \gamma (\gamma - 1)x_2^{ - 2}} \right] \hfill \\ \qquad\leqslant - C{r^\beta }x_2^\gamma ({r^{ - 2}} + x_2^{ - 2}) \hfill \\ \qquad\leqslant - C{r^{\beta - 1}}x_2^{\gamma - 1} \hfill \\ \qquad\leqslant - C{r^{\beta + \gamma - 2}} \hfill \\ \qquad\leqslant - C{(1 + |z|)^{\beta + \gamma - 2}}. \hfill \\ \end{gathered}

February 23, 2011

Some References for Geometric Evolution Equations

Filed under: Uncategorized — Ngô Quốc Anh @ 16:47

I found this syllabus for a course entitled Geometric Evolution Equations offered in SISSA. According to the material, all topics focus on the flow method in geometric analysis such as

  • Mean curvature flow;
  • Inverse mean curvature flow;
  • Willmore flow;
  • Ricci flow.

As can be seen from each topic there are several references that I am going to drop them here for convenience.

  1. Parabolic equations
    This topic focuses only on the theory of parabolic equations. To my opinion, the following two books are enough to fully understand this type of equations. 

    – Friedman, Avner. Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964.
    – Lieberman, Gary M.. Second order parabolic differential equations. World Scientific Publishing Co., Inc., River Edge, NJ, 1996.

  2. Mean curvature flow
    This topic focuses on the theory of mean curvature flow known as an example of a geometric flow of hypersurfaces in a Riemannian manifold. Intuitively, a family of surfaces evolves under mean curvature flow if the velocity of which a point on the surface moves is given by the mean curvature of the surface. 

    – Huisken, Gerhard. Flow by mean curvature of convex surfaces into spheres. J. Differential Geom.20 (1984), no. 1, 237-266. [here]
    – Huisken, Gerhard; Sinestrari, Carlo. Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math.183 (1999), no. 1 , 45-70. [here]
    – Zhu, Xi-Ping. Lectures on mean curvature flows. AMS/IP Studies in Advanced Mathematics, 32. American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002.

  3. Inverse mean curvature flow
    This topic focuses on the theory of inverse mean curvature flow. This is also an example of a geometric flow. Intuitively, a family of surfaces evolves under IMCF if the outward normal speed at which a point on the surface moves is given by the reciprocal of the mean curvature of the surface. 

    – Gerhardt,Claus. Flow of nonconvex hypersurfaces into spheres. J. Differential  Geom.32 (1990), no. 1, 299-314. [here]
    – Huisken, Gerhard; Ilmanen, Tom. The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differential Geom.59 (2001), no. 3 , 353-437. [here]

  4. Willmore flow
    This is the geometric flow corresponding to the Willmore energy; it is an L^2–gradient flow. This type of flow can be briefly described as follows. Over a surface M we can define the so-called Willmore energy of S 

    \displaystyle\mathcal{W} = \int_S H^2 dA-\int_S K dA

    where H is the mean curvature, K is the Gaussian curvature, and dA is the area form of S. Then we can define the Willmore flow as the following

    \displaystyle e[{\mathcal{M}}]=\frac{1}{2} \int_{\mathcal{M}} H^2\mathrm{d}A

    where H stands for the mean curvature of the manifold \mathcal{M}. Obviously, flow lines satisfy the differential equation

    \displaystyle\partial_t x(t) = -\nabla \mathcal{W}[x(t)]

    where x is a point belonging to the surface.

    This flow leads to an evolution problem in differential geometry: the surface \mathcal{M} is evolving in time to follow variations of steepest descent of the energy. Like surface diffusion (mathematics) it is a fourth-order flow, since the variation of the energy contains fourth derivatives.

    – Kuwert, Ernst; Schätzle, Reiner. The Willmore flow with small initial energy. J. Differential Geom.57 (2001), no. 3, 409-441. [here]
    – Kuwert, Ernst; Schätzle, Reiner. Gradient flow for the Willmore functional. Comm. Anal. Geom.10 (2002), no .  , 307-339. [here]
    – Simonett, Gieri. The Willmore flow near spheres. Differential Integral Equations14 (2001), no. 8, 1005-1014. [here]

  5. Ricci flow
    The Ricci flow is an intrinsic geometric flow (a process which deforms the metric of a Riemannian manifold) in this case in a manner formally analogous to the diffusion of heat, thereby smoothing out irregularities in the metric. The Ricci flow was first introduced by Richard Hamilton in 1981, and is also referred to as the Ricci-Hamilton flow. It plays an important role in Grigori Perelman‘s solution of the Poincaré conjecture, as well as in the proof of the Differentiable sphere theorem by Brendle and Schoen. 

    Given a Riemannian manifold with metric tensor g_{ij}, we can compute the Ricci tensor R_{ij}, which collects averages of sectional curvatures into a kind of “trace” of the Riemann curvature tensor. If we consider the metric tensor (and the associated Ricci tensor) to be functions of a variable which is usually called “time” (but which may have nothing to do with any physical time), then the Ricci flow may be defined by the geometric evolution equation

    \displaystyle\partial_t g_{ij}=-2 R_{ij}.

    The normalized Ricci flow makes sense for compact manifolds and is given by the equation

    \displaystyle\partial_t g_{ij}=-2 R_{ij} +\frac{2}{n} R_\mathrm{avg} g_{ij}

    where R_\mathrm{avg} is the average (mean) of the scalar curvature (which is obtained from the Ricci tensor by taking the trace) and n is the dimension of the manifold. This normalized equation preserves the volume of the metric.

    The factor of -2 is of little significance, since it can be changed to any nonzero real number by rescaling t. However the minus sign ensures that the Ricci flow is well defined for sufficiently small positive times; if the sign is changed then the Ricci flow would usually only be defined for small negative times. (This is similar to the way in which the heat equation can be run forwards in time, but not usually backwards in time.) Informally, the Ricci flow tends to expand negatively curved regions of the manifold, and contract positively curved regions.

    – Hamilton, Richard S.. Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (1982),  no. 2, 255-306. [here]
    – Hamilton, Richard S.. Four-manifolds with positive curvature operator. J. Differential Geom.24 (1986), no. 2, 153—179. [here]

February 19, 2011

Derivation of the relation between the nonlinear Schrodinger equation and its associated nonlinear elliptic equation

Filed under: Uncategorized — Ngô Quốc Anh @ 19:23

The main point of this entry is to derive the relation between the nonlinear Schrodinger equation and its associated nonlinear elliptic equation.

Let us start with the standing waves, \psi, for the following nonlinear Schrodinger equation in \mathbb R^n

\displaystyle -i\frac{\partial\psi}{\partial t}=\Delta \psi - \widetilde V(y)\psi + |\psi|^{p-1}\psi

where p>1.

By means of standing wave we look for \psi to be of the form

\psi(t,y)=e^{i\lambda t}u(y).

A simple calculation shows that

\displaystyle - i\frac{{\partial \psi }}{{\partial t}} = \lambda {e^{i\lambda t}}u(y).

Thus the NSL can be rewritten as follows

\displaystyle\lambda {e^{i\lambda t}}u(y) = {e^{i\lambda t}}\Delta u(y) - \widetilde V(y){e^{i\lambda t}}u(y) + |u(y){|^{p - 1}}u(y)

which is nothing but the following nonlinear elliptic equation

\displaystyle - \Delta u + V(y)u = |u{|^{p - 1}}u

where V(y)=\widetilde V(y)+\lambda.

In the literature, function u is usually called the amplitude. It is well-known that the amplitude is often assumed to be positive and to vanish at infinity, thus, it is reasonable to assume

u>0, \quad \mathop {\lim }\limits_{|y| \to + \infty } u(y) = 0.

In other words, our nonlinear elliptic equations reads as the following

\displaystyle - \Delta u + V(y)u = u^{p}, \quad y \in \mathbb R^n.

This kind of PDE has been extensively studied over years.

February 14, 2011

Stereographic projection, 3

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 18:07

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 not hard to see that

\displaystyle\frac{{\partial ({\xi _1},...,\xi _n,\xi_{n+1} )}}{{\partial ({x_1},...,x_n,\xi_{n+1})}} = \frac{2^n}{{{{\left( {1 + {{\left| x \right|}^2}} \right)}^{2n}}}}\det \left( {\begin{array}{*{20}{c}} {1 + {{\left| x \right|}^2} - 2x_1^2}&{ - 2{x_1}{x_2}}& \cdots &{ - 2{x_1}{x_n}} & 0 \\ { - 2{x_2}{x_1}}&{1 + {{\left| x \right|}^2} - 2x_2^2}& \cdots &{ - 2{x_2}{x_n}} & 0 \\ \vdots & \vdots & \ddots & \vdots \\ { - 2{x_n}{x_1}}&{ - 2{x_n}{x_2}}& \cdots &{1 + {{\left| x \right|}^2} - 2x_n^2} & 0\\ * & * & \cdots & * & 1 \end{array}} \right).

(more…)

February 11, 2011

The implicit function theorem: An ODE example

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 23:35

We want to continue the series of notes involving some applications of  the implicit function theorem. As in the previous note, here we consider the solvability of the following ODE

x''+\mu x+f(x)=0, \quad J= [0,1]

with the following boundary conditions

x(0)=0=x(1).

We assume that f \in C^1(\mathbb R) and f(0)=0. For the sake of convenience, let us recall the standard implicit function theorem

Theorem (implicit function theorem). Let X, Y, Z be Banachspaces, U\subset X and V\subset Y neighbourhoods of x_0 and y_0 respectively, F: U\times V\to Z continuous and continuously differentiable with respect to y. Suppose also that

F(x_o, y_o) = 0,\quad F_y^{-1}(x_o,y_o) \in L(Z, Y).

Then  there exist balls \overline B_r(x_o) \subset U, \overline B_r (y_o) \subset V and exactly one map T: B_r(x_o) \to B_r (y_o) such that

Tx_o = y_o and F(x, Tx) = 0 on B_r(x_o).

This map T is continuous.

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February 8, 2011

Minimal Surfaces: The mean curvature condition

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 1:21

Suppose that \vec x(u,v), (u,v) \in I_1 \times I_2 is the surface with minimal area among those whose boundary coincides with that of \vec x. Let t(u,v) be any smooth function on I_1 \times I_2 such that it vanishes on the boundary of \vec x

Consider the following family of regular parametrized surfaces

{\vec x}^\varepsilon = \vec x(u,v)+\varepsilon t(u,v)N, \quad -\lambda < \varepsilon < \lambda

for some small \lambda. These surfaces can be thought of as being obtained by varying \vec x(u,v) along its normals, while fixing its boundary. Note that

\displaystyle {E^\varepsilon } = \langle \vec x_u^\varepsilon ,\vec x_u^\varepsilon \rangle = E + 2\left\langle {{{\vec x}_u},\varepsilon \left( {\frac{{\partial t}}{{\partial u}}N + t\frac{{\partial N}}{{\partial u}}} \right)} \right\rangle + {\varepsilon ^2}{\left\| {\frac{{\partial t}}{{\partial u}}N + t\frac{{\partial N}}{{\partial u}}} \right\|^2}

and

\displaystyle \begin{gathered} {F^\varepsilon } = \langle \vec x_u^\varepsilon ,\vec x_v^\varepsilon \rangle \hfill \\ \quad\;= F + \left\langle {{{\vec x}_u},\varepsilon \left( {\frac{{\partial t}}{{\partial v}}N + t\frac{{\partial N}}{{\partial v}}} \right)} \right\rangle + \left\langle {{{\vec x}_v},\varepsilon \left( {\frac{{\partial t}}{{\partial u}}N + t\frac{{\partial N}}{{\partial u}}} \right)} \right\rangle + {\varepsilon ^2}\left\langle {\frac{{\partial t}}{{\partial u}}N + t\frac{{\partial N}}{{\partial u}},\frac{{\partial t}}{{\partial v}}N + t\frac{{\partial N}}{{\partial v}}} \right\rangle \hfill \\ \end{gathered}

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February 3, 2011

Components of a tensor under a change of coordinates

Filed under: Riemannian geometry — Ngô Quốc Anh @ 19:29

During the Cosmology class several days ago, the lecturer told us the following interesting thing that would be useful for beginner studying Riemannian geometry.

Let us consider a covariant vector, known as (1,0)-tensor, v_\alpha dx^\alpha. The word covariant says that the components, v_\alpha, vary in the same “direction” as the change of coordinates. To be precise, under a change of coordinates

\displaystyle x^\alpha \mapsto x^{\overline \alpha}

we have

{v_\alpha }d{x^\alpha } = \underbrace {{v_\alpha }\frac{{\partial {x^\alpha }}}{{\partial {x^{\overline \alpha }}}}}_{{v_{\overline \alpha }}}d{x^{\overline \alpha }}.

Now let us consider a contravariant vector, also known as (0,1)-tensor, v^\alpha \frac{\partial}{\partial x^\alpha}. The word contravariant says that the components, v_\alpha, vary in the opposite “direction” as the change of coordinates. To be precise, under a change of coordinates as above, we obtain

\displaystyle {v^\alpha }\frac{\partial }{{\partial {x^\alpha }}} = {v^\alpha }\frac{{\frac{\partial }{{\partial {x^\alpha }}}}}{{\frac{\partial }{{\partial {x^{\overline \alpha }}}}}}\frac{\partial }{{\partial {x^{\overline \alpha }}}} = \underbrace {{v^\alpha }\frac{{\partial {x^{\overline \alpha }}}}{{\partial {x^\alpha }}}}_{{v^{\overline \alpha }}}\frac{\partial }{{\partial {x^{\overline \alpha }}}}.

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