Ngô Quốc Anh

May 30, 2010

Co-area formula for gradient, 2

Filed under: Giải Tích 6 (MA5205) — Tags: — Ngô Quốc Anh @ 20:29

Let us recall the following result

Theorem. Let \Omega \subset \mathbb R^n be an open set and u \in \mathcal D(\Omega). If u \geqslant 0 then for any 1 \leqslant p<\infty, we have

\displaystyle\int_\Omega|\nabla u|^p dx =\int_0^M \left(\int_{\{u=t\}}|\nabla u|^{p-1}d\sigma\right)dt

where M=\sup u over \overline \Omega.

The above result has been proven in this entry. If we chose p=0 then we would have

\displaystyle\int_\Omega dx =\int_0^M \left(\int_{\{u=t\}}\frac{1}{|\nabla u|}d\sigma\right)dt.

If we now replace \Omega=\{u>t\}, we get

\displaystyle\int_{\{u>t\}}dx =\int_t^M \left(\int_{\{u=t\}}\frac{1}{|\nabla u|}d\sigma\right)dt.

By differentiating with respect to t, we arrive at

\displaystyle \frac{d}{dt}\int_{\{u>t\}}dx =-\int_{\{u=t\}}\frac{1}{|\nabla u|}d\sigma.

In this entry, we shall prove the foregoing identity is actually true. This is the second interesting formula I have mentioned before. The proof, of course again, is based on a clever choice of test function.

(more…)

Advertisements

May 28, 2010

Concentration-Compactness principle, I

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 7:32

In this entry, we talk about the Concentration-Compactness Principle discovered by P.L. Lions [here].

Theorem (Lions). Suppose that \{u_m\} is a bounded sequence in W^{1,p}(\mathbb R^n), 2<p<n and let

\displaystyle {\rho _m} = {\left| {{u_m}} \right|^p}

with

\displaystyle\int_{{\mathbb{R}^n}} {{\rho _m}} = \lambda

for all m. Then there exists a subsequence \rho_{m_k} satisfying one of the three following possibilities

  1. (Compactness) There exists a sequence \{y^k\} in \mathbb R^n such that \rho_{m_k} is tight, that is, for every \varepsilon>0 there exists 0<R<\infty such that

    \displaystyle \int_{{B_R}({y^k})} {{\rho _{{m_k}}}} \geqslant \lambda - \varepsilon .

  2. (Vanishing)

    \displaystyle \mathop {\lim }\limits_{k \to \infty } \mathop {\sup }\limits_{y \in {\mathbb{R}^n}} \int_{{B_R}(y)} {{\rho _{{m_k}}}} = 0, \quad \forall R > 0.

  3. (Dichotomy) There exists \alpha \in (0,\lambda) such that for all \varepsilon >0, there exist k_0 \geqslant 1, bounded sequences \{u_k^1\} and \{ u_k^2\} in W^{1,p}(\mathbb R^n) satisfying for k \geqslant k_0

    \displaystyle\int_{{\mathbb{R}^n}} {{{\left( {{u_{{m_k}}} - u_k^1 - u_k^2} \right)}^q}} \leqslant {\delta _q}(\varepsilon ), \quad \forall p \leqslant q < {p^*} = \frac{{np}}{{n - p}}

    with \delta_q(\varepsilon)\to 0 as \varepsilon\to 0 and

    \displaystyle\left| {\int_{{\mathbb{R}^n}} {{{\left| {u_k^1} \right|}^p}dx} - \alpha } \right| < \varepsilon

    and

    \displaystyle\left| {\int_{{\mathbb{R}^n}} {{{\left| {u_k^2} \right|}^p}dx} - (\lambda - \alpha )} \right| < \varepsilon

    and

    \displaystyle {\rm dist}\left( {{\rm supp} \; u_k^1,{\rm supp} \; u_k^2} \right) \to 0, \quad k \to \infty

    and

    \displaystyle\mathop {\lim \inf }\limits_{k \to \infty } {\int_{{\mathbb{R}^n}} {{{\left| {\nabla {u_{{m_k}}}} \right|}^p} - {{\left| {\nabla u_k^1} \right|}^p} - \left| {\nabla u_k^2} \right|} ^p} \geqslant 0.

(more…)

May 27, 2010

A Simple Approach to the Hardy and Rellich inequalities

Filed under: Giải Tích 6 (MA5205) — Tags: , — Ngô Quốc Anh @ 16:07

The classical Hardy inequality in \mathbb R^n, n \geqslant 3, is stated as follows

Theorem (Hardy’s inequality). Let u \in \mathcal D^{1,2}(\mathbb R^n) with n \geqslant 3. Then

\displaystyle\frac{{{u^2}}}{{{{\left| x \right|}^2}}} \in {L^1}({\mathbb{R}^n})

and

\displaystyle {\left( {\frac{{n - 2}}{2}} \right)^2}\int_{{\mathbb{R}^n}} {\frac{{{u^2}}}{{{{\left| x \right|}^2}}}dx} \leqslant \int_{{\mathbb{R}^n}} {{{\left| {\nabla u} \right|}^2}dx}.

The constant {\left( {\frac{{n - 2}}{2}} \right)^2} is the best possible constant.

I suddenly found a very simple proof due to E. Mitidieri [here].

(more…)

May 26, 2010

Coupled Fixed-Point Principle

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 20:03

I recently learned from a paper due to Michael Holst et al. published in Comm. Math. Phys. in 2009 [here] the following two fixed-point principles that may be useful elsewhere.

Theorem (Coupled Fixed-Point Principle A). Let X and Y be Banach spaces, and let Z be a Banach space with compact embedding X: \hookrightarrow Z. Let U \subset Z be a non-empty, convex, closed, bounded subset, and let

S : U \to \mathcal R(S) \subset Y,\quad T : U\times \mathcal R(S)\to U \cap X,

be continuous maps. Then there exist \varphi \in U \cap X and w \in \mathcal R(S) such that

\varphi = T (\varphi,w) and w = S(\varphi).

The proof will be through a standard variation of the Schauder Fixed-PointTheorem.

  1. Construction of a non-empty, convex, closed, bounded subset U \subset Z.
  2. Continuity of a mapping G : U\subset Z \to U \cap X \subset X.
  3. Compactness of a mapping F : U \subset Z \to U \subset Z.
  4. Invoking the Schauder Theorem.

In particular, we have the following variant which is useful in practice.

Theorem (Coupled Fixed-Point Principle B). Let X and Y be Banach spaces, and let Z be a real ordered Banach space having the compact embedding X: \hookrightarrow Z. Let [\varphi^-, \varphi^+] \subset Z be a nonempty interval which is closed in the topology of Z, and set

U = [\varphi^-, \varphi^+] \cap \overline B_M \subset Z,

where \overline B_M is the closed ball of finite radius M >0 in Z about the origin. Assume U is nonempty, and let the maps

S : U \to \mathcal R(S) \subset Y,\quad T : U\times \mathcal R(S)\to U \cap X,

be continuous maps. Then there exist \varphi \in U \cap X and w \in \mathcal R(S) such that

\varphi = T (\varphi,w) and w = S(\varphi).

The proof is just verification of the first theorem. For a proof of all, I refer the reader to the paper mentioned above.

May 24, 2010

Co-area formula for gradient

Filed under: Giải Tích 6 (MA5205) — Tags: — Ngô Quốc Anh @ 18:55

I found two interesting formulas related to co-area formula while reading some tricks done by Talenti regarding to the best constant of the Sobolev inequality. The first result is to derive a representation of

\displaystyle \int_\Omega |\nabla u|^pdx

and the second result is to deal with differentiation of level sets. Having all these stuffs, I will derive a very short and beautiful proof concerning the lower bound of \int \exp(u) dx where u, a positive solution to the following PDE

\displaystyle-\Delta u = e^u

This proof I firstly learned from a paper published in Duke Math. J. in 1991 by W. Cheng and C. Li [here].

Co-area formula. Suppose that \Omega is an open set in \mathbb R^n, and u is a real-valued Lipschitz function on \Omega. Then, for an integrable function g

\displaystyle\int_\Omega g(x) |\nabla u(x)| dx = \int_{-\infty}^\infty \left(\int_{\{u=t\}}g(x) dH_{n-1}(x)\right)dt

where H^{n-1} is the (n-1)-dimensional Hausdorff measure.

The Sard theorem. Let f :\mathbb{R}^n \rightarrow \mathbb{R}^m be C^k, k times continuously differentiable, where k \geqslant \max\{n-m+1, 1\}. Let X be the critical set of f, the set of points x in \mathbb R^n at which the Jacobian matrix of f has {\rm rank} < m. Then f(X) has Lebesgue measure 0 in \mathbb R^m.

The Sard theorem has some useful applications. For example, if u \in \mathcal D(\Omega) the space of test functions where \Omega \subset \mathbb R^n, then for almost every t in the range of u, we have that |\nabla u|\ne 0 on the level set \{u=t\}. Thus that level set will be an (n-1)-dimensional surface. Furthermore

\{u=t\}=\partial \{u>t\}

and

|\{u=t\}|=0.

Theorem. Let \Omega \subset \mathbb R^n be an open set and u \in \mathcal D(\Omega). If u \geqslant 0 then for any 1 \leqslant p<\infty, we have

\displaystyle\int_\Omega|\nabla u|^p dx =\int_0^M \left(\int_{\{u=t\}}|\nabla u|^{p-1}d\sigma\right)dt

where M=\sup u over \overline \Omega.

(more…)

May 22, 2010

The third and fouth fundamental results in the calculus of variation

Filed under: Giải tích 8 (MA5206), PDEs — Ngô Quốc Anh @ 17:56

Followed by this entry where the following questions have been discussed

  1. Boundedness implies weakly convergence: If E is a reflexive Banach space and \{x_n\}_n \subset E is a bounded sequence. Then up to a subsequence x_n converges weakly to some x in X.
  2. Weakly convergence becomes strongly convergence via compact operator: A compact operator C : E \to X between Banach spaces maps every weakly convergent sequence in E into one that converges strongly in X.

Now I shall discuss more results which appear frequently in the calculus of variation.

Let us recall over a minifold M, the Sobolev norm H^1(M) (or W^{1,2}(M)) is defined by

\displaystyle \|u\|_{H^1}^2=\|u\|_{L^2}^2+\|\nabla u\|_{L^2}^2.

It is immediately to see that if u_n \to u strongly in H^1, i.e. \|u_n-u\|_{H^1}\to 0 then u_n \to u strongly in L^2 and \nabla u_n \to \nabla u strongly in L^2.

It turns out to discuss what happen to weakly convergence. Actually, we shall prove the following important result, called the third fundamental result.

Weakly convergence in Sobolev spaces implies weakly convergence in L^p spaces. We assume u_n \rightharpoonup u in H^1. We shall prove both u_n and \nabla u_n converge weakly to u and \nabla u in L^2, respectively.

By the principle of uniform boundedness, any weakly convergence sequence is bounded in the norm. Consequently, \{u_n\}_n and \{\nabla u_n\}_n are bounded in L^2. By the weak compactness of balls in L^2, there is a subsequence n_k such that

\displaystyle u_{n_k} \rightharpoonup v, \quad \nabla u_{n_k} \rightharpoonup w

in L^2 (i.e., both v, w are in L^2). Since the weak convergence in L^2 implies the convergence in \mathcal D' the dual space of \mathcal D-the space of test functions. It follows that w = \nabla v and, hence, v \in H^1. It follows that u\equiv v and thus

\displaystyle u_{n_k} \rightharpoonup u, \quad \nabla u_{n_k} \rightharpoonup \nabla u

in L^2 as desired.

Now we consider the reverse case. We shall prove the following

Weakly convergence in L^p spaces plus the boundedness implies weakly convergence in Sobolev spaces. We assume u_n \rightharpoonup u \in L^2 in L^2 and \|u_n\|_{H^1} is bounded. We shall prove that u \in H^1 and u_n \rightharpoonup u in H^1.

Since \{u_n\} is bounded in H^1, by the first fundamental result, u_n \rightharpoonup v in H^1 for some v \in H^1. By the third fundamental result above, u_n \rightharpoonup v in L^2. It follows from the uniqueness of weak limit that u \equiv v which implies u \in H^1.

In order to prove u_n \rightharpoonup u in H^1, we shall use the following result whose proof is based on the simple contradiction argument.

Let X be a topological space. A sequence \{x_n\} \subset X  converges to x \in X (in the topological of X) if and only if any subsequence of \{x_n\} contains a sub-subsequence that converges to x.

Let us pick a particular subsequence of \{u_n\} and rename it back to \{u_n\} for simplicity. It suffices to prove that \{u_n\} contains a subsequence that converges to u weakly in H^1. Followed the proof of the third fundamental result, there is a subsequence of \{u_n\} and a function v \in H^1 such that

\displaystyle u_{n_k} \rightharpoonup v, \quad \nabla u_{n_k} \rightharpoonup v

both in L^2. It follows from the definition of weakly convergence in H^1 that in fact we get

\displaystyle u_{n_k} \rightharpoonup v

in H^1. The reason is the following:

\displaystyle (u_{n_k},\varphi)_{L^2}+(\nabla u_{n_k},\nabla\varphi)_{L^2} \to (v,\varphi)_{L^2} + (\nabla v, \varphi)_{L^2}

for any \varphi \in H^1. Having this and the fact that weak limit is unique we deduce that u \equiv v. The latter now implies

u_{n_k} \rightharpoonup u

in H^1. The proof follows.

See also: Two fundamental results in the calculus of variation

May 21, 2010

How to remember the definition of sup- and super-solutions?

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 17:10

I am frequently confused the definition of sup- and super-solutions so yesterday I tried to figure out a way to remember those things. Fortunately, I think I got a very simple way to remember. The point is, just denote by \underline u and \overline u the sup- and super-solutions respectively to the very simple PDE

-\Delta u = f(x,u),

which one is true

\displaystyle -\Delta \overline u \leqslant f(x,\overline u)

or

\displaystyle -\Delta \overline u \geqslant f(x,\overline u)

and similarly to \underline u.

Let us consider a general case. Assume we are working with a general second-order elliptic operator in non-divergence form, say

\displaystyle L=-a^{ij}\partial_i\partial_j + b^k\partial_k + c

where a^{ij} are positive coefficients.  Besides, coefficients of L are assumed to satisfy several conditions like symmetry, etc. but it is not considered here.

Concerning the following PDE

L(u)=f(x,u),

we say

function \overline u (resp. \underline u) is said to be a super-solution (resp. sub-solution) to the PDE if L(\overline u) \geqslant f(x,\overline u) (resp. L(\underline u) \leqslant f(x,\underline u)).

Observe that ^{ij} is positive lets us think that L has positive spectrum. So once \overline u is a super-solution, the inequality should be {\rm LHS} \geqslant {\rm RHS} where the left hand side should be an operator with positive spectrum. Similarly, concerning the sub-solutions, we need {\rm LHS} \leqslant {\rm RHS}.

Let us go back to the Laplacian operator \Delta = \partial^i\partial_i. So \Delta has negative spectrum, this yields

\displaystyle \Delta \overline u \leqslant -f(x,\overline u)

and

\displaystyle \Delta \underline u \geqslant -f(x,\underline u).

Equivalently,

\displaystyle -\Delta \overline u \geqslant f(x,\overline u)

and

\displaystyle -\Delta \underline u \leqslant f(x,\underline u)

since -\Delta has positive spectrum.

May 20, 2010

Fractional Laplacian in R^N

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 17:29

Half-Laplacian has been discussed in this entry using harmonic extension. Today, we derive a more general form, called fractional laplacian denoted by (-\Delta)^s, for a function f : \mathbb R^n \to \mathbb R where the parameter s is a real number between 0 and 1 and n \geqslant 3.

It is worth recalling the following fact known as Newtonian potential in theory of PDEs. The Newtonian potential u of a compactly supported integrable function f, i.e. f \in L_{loc}^1(\mathbb R^n), is defined as the convolution

\displaystyle u(x) = \Gamma * f(x) = \int_{\mathbb{R}^n} \Gamma(x-y)f(y)dy

where the Newtonian kernel \Gamma in dimension n is defined by

\displaystyle\Gamma(x) = \begin{cases} \frac{1}{2}\left| x \right| & n=1 \\ \frac{1}{2\pi} \log{ | x | } & n=2 \\ \frac{1}{n(2-n)\omega_n} | x | ^{2-n} & n>2. \end{cases} .

Here \omega_d is the volume of the unit in \mathbb R^n. Coefficients in \Gamma(x), denoted by \frac{1}{C_n}, are usually called normalization constants. The Newtonian potential u of f is a solution of the Poisson equation

\Delta u = f.

In case n \geqslant 3, u is actually the Riesz potential

\displaystyle u(x)=(I_2f)(x)=\frac{1}{C_{n,2}} \int_{\mathbb{R}^n}\frac{f(y)}{|x-y|^{n-2}}dy

with \alpha=2 where the Riesz potential is defined by

\displaystyle (I_{\alpha}f) (x)= \frac{1}{C_{n,\alpha}} \int_{{\mathbb{R}}^n} \frac{f(y)}{| x - y |^{n-\alpha}} dy

where C_{n,s} is some normalization constant given by

\displaystyle C_{n,s} = \pi^{n/2}2^s\frac{\Gamma\left(\frac{s}{2}\right)}{\Gamma\left(\frac{n-s}{2}\right)}.

Observe that at least formally the Riezs potentials I_\alpha verify the following rule

\displaystyle \Delta I_{\alpha+2} = -I_\alpha, \quad 0 \leqslant \alpha \leqslant n

which implies

\displaystyle \Delta^2 I_{\alpha+2} = -\Delta I_\alpha=I_{\alpha-2}, \quad 2 \leqslant \alpha \leqslant n-2.

Thus if we set \alpha to be 0 (in fact, this is impossible) we get

\displaystyle \Delta f = \Delta^2 u=\Delta^2 (I_2f)=I_{-2}f

which helps us to write down

\displaystyle \Delta f=\frac{1}{C_{n,2}} \int_{\mathbb{R}^n}\frac{f(y)}{|x-y|^{n+2}}dy.

Thus we wish to define the fractional Laplacian (-\Delta)^s as follows

\displaystyle (-\Delta )^s f(x) = \frac{1}{C_{n,s}} \int_{\mathbb{R}^n} \frac{-f(\xi)}{|x-\xi|^{n+2s}}d\xi.

The above fractional Laplacian is also often called the Riesz fractional derivative [here]. In a paper entitled “An Extension Problem Related to the Fractional Laplacian” due to Luis Caffarelli et al. [here] published in Comm. Partial Differential Equations in 2007, the fractional Laplacian can also be defined using

\displaystyle (-\Delta )^s f(x) = \frac{1}{C_{n,s}} \int_{\mathbb{R}^n} \frac{f(x)-f(\xi)}{|x-\xi|^{n+2s}}d\xi.

Let us now study the terminology of weak solution to the following semilinear elliptic equation in the whole space

\displaystyle (-\Delta)^\frac{\alpha}{2}u=u^\frac{n+\alpha}{n-\alpha}.

By a weak solution we mean a function u \in H^\frac{\alpha}{2}(\mathbb R^n) such that

\displaystyle\int_{\mathbb R^n} (-\Delta)^\frac{\alpha}{4}u (-\Delta)^\frac{\alpha}{4} \varphi dx = \int_{\mathbb R^n} u^\frac{n+\alpha}{n-\alpha} \varphi dx

for any positive test function \varphi in the distribution sense. I will back to this stuff once I finish introducing the fractional Laplacian via pseudo-differential operators.

See also: Half-Laplacian in \mathbb R^n

May 16, 2010

Conformal Changes of Riemannian Metrics

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 5:54

I guess I will use the relation between curvature tensors of metrics lying in a conformal class frequently so I decide to post something related to this stuff which may be helpful and we can use later. Actually, I have used it when we proved conformal Laplacian operator is invariant. Let us briefly recall some terminologies

Definition (conformal). Two pseudo-Riemannian metrics g and \widetilde g on a manifold M are said to be

  1. (pointwise) conformal if there exists a C^\infty function f on M such that

    \displaystyle \widetilde g=e^{2f}g;

  2. conformally equivalent if there exists a diffeomorphism \alpha of M such that \alpha^* \widetilde g and g are pointwise conformal.

Note that, if g and \widetilde g are conformally equivalent, then \alpha is an isometry from e^{2f}g onto \widetilde g. So we will only study below the case \widetilde g = e^{2f}g. Our aim is to compare Riemann curvature, Scalar curvature, Ricci curvature,… of g and \widetilde g.

Definition (the Kulkarni–Nomizu product). This product \odot is defined for two (0,2)-tensors and gives as a result a (0,4)-tensor. Precisely,

\displaystyle \alpha \odot \beta (X_1,X_2,X_3,X_4)=\alpha (X_1,X_3)\beta (X_2,X_4)+\alpha (X_2,X_4)\beta (X_1,X_3)-\alpha (X_1,X_4)\beta (X_2,X_3)-\alpha (X_2,X_3)\beta (X_1,X_4)

or

\displaystyle {(\alpha \odot \beta )_{ijkl}} = {\alpha _{il}}{\beta _{jk}} + {\alpha _{jk}}{\beta _{il}} - {\alpha _{ik}}{\beta _{jl}} - {\alpha _{jl}}{\beta _{ik}}.

Levi-Civita connection. On (M,g), the Levi-Civita connection \nabla is an affine connection which is torsion free

\displaystyle \nabla_XY+\nabla_YX=[X,Y]

and satisfies the rule

\displaystyle X(g(Y,Z))=g(\nabla_X Y,Z) + g(Y, \nabla_X Z)

for any vector fields X,Y,Z. We now have

\displaystyle {\widetilde\nabla _X}Y = {\nabla _X}Y + X(f)Y + Y(f)X - g(X,Y) {\rm grad}f.

Weyl tensor. This tensor is defined to be

\displaystyle W = R - \frac{1}{{n - 2}}\left( {{\rm Ric} - \frac{{\rm Scal}}{n}g} \right) \odot g - \frac{{\rm Scal}}{{2n(n - 1)}}g \odot g.

Thus we have the rule

\displaystyle\widetilde W =W.

Ricci tensor. This is a (2,0)-tensor defined by

\displaystyle {\rm Ric}(X,Y) = {\rm Trace}( x \to R(x, X)Y).

In local coordinates, it has the form

\displaystyle {\rm Ric} = R_{ij} dx^i \otimes dx^j.

So we have the following rule

\displaystyle\widetilde{\rm Ric} = {\rm Ric} - (n - 2)({\rm Hess} f -{\rm grad}f \otimes {\rm grad}f) + (\Delta f - (n - 2)|{\rm grad}f|^2)g.

Traceless Ricci tensor. This tensor is defined by

\displaystyle\displaystyle {Z_{ij}} = {R_{ij}} - \frac{1}{n}{\rm Scal}{g_{ij}}.

A simple calculation shows that its trace, g^{ij}Z_{ij}, equals zero. So

\displaystyle\widetilde Z = Z - (n - 2)\left( {{\rm Hess}f - {\rm grad}f \otimes {\rm grad}f} \right) - \frac{{n - 2}}{n}\left( {\Delta f + |{\rm grad}f|^2} \right)g.

Scalar curvature. This (2,0) tensor is defined to be the trace of Ricci tensor, that is

\displaystyle {\rm Scal} = {\rm Trace}( {\rm Ric}) = g^{jk}{\rm Ric}_{jk}.

So

\displaystyle\widetilde{\rm Scal} = {e^{ - 2f}}\left[ {{\rm Scal} + 2(n - 1)\Delta f - (n - 2)(n - 1)|{\rm grad} f{|^2}} \right].

In practice, this conformal change is not useful, we usually use the following conformal change

\displaystyle \widetilde g=f^\frac{4}{n-2}g.

With this, we simply have

\displaystyle - \Delta f + \frac{{n - 2}}{4(n - 1)}{\rm Scal}f = \frac{{n - 2}}{4(n - 1)}\widetilde{{\rm Scal}}{f^{\frac{{n + 2}}{{n - 2}}}}

or

\displaystyle \widetilde{{\rm Scal}} = {f^{ - \frac{{n + 2}}{{n - 2}}}}\left[ { - \frac{4(n - 1)}{n - 2}\Delta f + {\rm Scal}f} \right].

Riemann curvature tensor. This (1,3) tensor is defined to be

\displaystyle R(X,Y)Z = \nabla_X \nabla_YZ - \nabla_Y \nabla_XZ - \nabla_{[X,Y]}Z.

In local coordinates, we get

\displaystyle R = R_{ikl}^j\dfrac{\partial }{{\partial {x^j}}} \otimes d{x^i} \otimes d{x^k} \otimes d{x^l}.

So

\displaystyle \widetilde R = {e^{2f}}\left[ {R - g\odot\left( {{\rm Hess}f - {\rm grad}f \otimes {\rm grad}f + \frac{1}{2}|{\rm grad}f{|^2}g} \right)} \right].

Volume element. This, d{\rm vol}_g, is the unique density such that, for any orthonormal basis (X_i) of T_XM,

\displaystyle d{\rm vol}_g(X_1,...,X_n)=1.

In local coordinates,

\displaystyle d{\rm vol}_g = \sqrt{|g|} dx^1\wedge \dots \wedge dx^n.

So

\displaystyle d{\rm vol}_{\widetilde g} = e^{nf}d{\rm vol}_g.

Hodge operator on p-forms (if M is oriented). The Hodge star operator on an oriented inner product space V is a linear operator on the exterior algebra of V, interchanging the subspaces of k-vectors and n-k-vectors where n = \dim V, for 0 \leqslant k \leqslant n. It has the following property, which defines it completely: given an oriented orthonormal basis e_1,e_2,\dots,e_n we have

\displaystyle *(e_{i_1} \wedge e_{i_2}\wedge \cdots \wedge e_{i_k})= e_{i_{k+1}} \wedge e_{i_{k+2}} \wedge \cdots \wedge e_{i_n}.

One can repeat the construction above for each cotangent space of an n-dimensional oriented Riemannian or pseudo-Riemannian manifold, and get the Hodge dual n-k-form, of a k-form. The Hodge star then induces an L^2-norm inner product on the differential forms on the manifold. One writes

\displaystyle (\eta,\zeta)=\int_M \eta\wedge *\zeta

for the inner product of sections \eta and \zeta of \Lambda^k(M). (The set of sections is frequently denoted as \Omega^k(M) = \Gamma(\Lambda^k(M)). Elements of \Omega^k(M) are called exterior k-forms). For example, for a positively oriented orthogonal cofram \{\omega^i\}_1^n, one has

\displaystyle *(\omega^1 \wedge \cdots \wedge \omega^p)=\omega^{p+1}\wedge \cdots \wedge \omega^n.

So

\displaystyle {*_{\widetilde g}} = {e^{(n - 2p)f}}{*_g}.

Codifferential on p-forms. This notion \delta is usually defined through the exterior derivative

\displaystyle d : \Omega^p(M) \to \Omega^{p+1}(M)

by the following rule (also called the formal adjoint of exterior derivative)

\displaystyle\langle \eta,\delta \zeta\rangle = \langle d\eta,\zeta\rangle,

i.e.

\displaystyle \delta : \Omega^p(M) \to \Omega^{p-1}(M).

In other words, for a p-form \beta,

\displaystyle \delta \beta = {( - 1)^{np + n + 1}}*d*\beta .

So

\displaystyle\widetilde\delta \beta = {e^{ - 2f}}\left[ {\delta \beta - (n - 2p){\iota_{{\rm grad}f}}\beta } \right]

where \iota denotes the interior product (the contraction of a differential form with a vector field).

(pseudo-) Laplacian on p-forms. This is known as the Hodge Laplacian and also known as the Laplace–de Rham operator. It is defined by

\displaystyle\Delta= d\delta+\delta d.

An important property of the Hodge Laplacian is that it commutes with the * operator, i.e.

\Delta * = * \Delta.

So

\displaystyle\widetilde\Delta \alpha = {e^{ - 2f}}\left[ {\Delta \alpha - (n - 2p)d({\iota_{{\rm grad}f}}\alpha ) - (n - 2p - 2){\iota_{{\rm grad}f}}d\alpha + 2(n - 2p){\rm grad}f \wedge {\iota_{{\rm grad}f}}\alpha - 2{\rm grad}f \wedge \delta \alpha } \right].

See also: Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, 1987.

May 15, 2010

Two fundamental results in the calculus of variation

Filed under: Giải tích 8 (MA5206), PDEs — Ngô Quốc Anh @ 20:28

I suddenly think that I should post this entry ‘cos sometimes I don’t remember these stuffs. These results appear frequently in solving PDEs especially when using the direct method. For example, the simplest case is the following eigenvalue problem

\displaystyle -{\rm div}(|\nabla u|^{p-2}\nabla u)=\lambda |u|^{p-2}u

over a bounded domain \Omega \subset \mathbb R^n with Dirichlet boundary condition. We assume p<n. Our aim is to show the existence of the first eigenvalue \lambda_1>0. Obviously, our problem is to solve the following optimization

\displaystyle\mathop {\inf }\limits_{u \in W^{1,2}_0(\Omega)} \left\{ {\int_\Omega {|\nabla u{|^p}dx} :\int_\Omega {|u{|^p}dx} = 1} \right\}.

The direct method says that we firstly select a minimizing sequence, say \{u_n\}_n, then we need to prove \{u_n\} is convergent. There are two steps in the above argument which lead to this entry. Our first claim is the following.

Boundedness implies weakly convergence. The first result says that

If E is a reflexive Banach space and \{x_n\}_n \subset E is a bounded sequence. Then up to a subsequence x_n converges weakly to some x in X.

The proof of this claim can be found in a book due to Brezis (Theorem III.27). Interestingly, its converse also holds by the Eberlein-Šmulian theorem.

Theorem (Eberlein-Šmulian). Suppose E is a Banach space such that every bounded sequence \{x_n\}_n contains a weakly convergent subsequence. Then E is reflexive.

There was an elementary proof of this theorem. We refer the reader to a paper due to Whitley [here]. Let us get back to our optimization problem. Once we have a minimizing sequence \{u_n\}_n \subset W^{1,p}_0(\Omega) it is clear to see that \{u_n\}_n is bounded in W^{1,p}_0(\Omega) since

\displaystyle\int_\Omega {|{u_n}{|^p}dx} = 1

and

\displaystyle\int_\Omega {|\nabla {u_n}{|^p}dx} \leqslant C, \quad \forall n.

By using the first claim, u_n converges weakly to some u \in W^{1,p}_0(\Omega). It is worth noticing that by saying u_n \rightharpoonup u in W^{1,p}_0(\Omega) we mean u_n \rightharpoonup u in L^p(\Omega) and Du_n \rightharpoonup Du in L^p(\Omega, \mathbb R^n). Now we need further argument

Weakly convergence becomes strongly convergence via compact operator. This second result says that

A compact operator C : E \to X between Banach spaces maps every weakly convergent sequence in E into one that converges strongly in X.

The proof of this relies on the contradiction argument and the fact that once a sequence converges strongly to some limit, this limit is unique. However, the converse is no long true. For example, by the Schur theorem, a sequence \{x_n\} in \ell^1 converges weakly, it also converges strongly. We take C to be the identity I in \ell^1. Since \ell^1 has infinity dimensional, by using the Riesz theorem, I cannot be compact.

Using this claim we deduce that u_n converges strongly to u in L^q for any 1\leqslant q<p^\star=\frac{np}{n-p}. By using the Minkowski and Holder inequalities we can show that u satisfies the constraint. It now follows from the weakly lower semi-continuous of norm that u indeed satisfies the equation. The proof follows.

Older Posts »

Blog at WordPress.com.