Concentration-Compactness principle, I

May 28, 2010

Concentration-Compactness principle, I

Filed under: PDEs — Tags: Concentration-Compactness — 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

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

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

(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

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

We now briefly explain how the principle of loss of mass at infinity [here] leads to the above Theorem. Let $\{u_m\}$ be a bounded sequence in $W^{1,2}(\mathbb R^n)$ such that

$\displaystyle 0 < \lambda = \int_{{\mathbb{R}^n}} {{{\left| {{u_m}(x)} \right|}^p}dx} , \quad 2 < p < 2^\star$

for all $m \geqslant 1$ . We may assume that

$\displaystyle {u_m} \rightharpoonup u$

in $W^{1,2}(\mathbb R^n)$ . According to the principle of loss of mass at infinity, we have

$\displaystyle\lambda = \int_{{\mathbb{R}^n}} {{{\left| {u(x)} \right|}^p}dx} + {\alpha _\infty }$ .

We now distinguish three cases

$\alpha _\infty=\lambda$ and thus $u \equiv 0$ (almost everywhere)
$\alpha _\infty=0$ and thus
$\displaystyle\lambda = \int_{{\mathbb{R}^n}} {{{\left| {u(x)} \right|}^p}dx}$
and
$\displaystyle\int_{{\mathbb{R}^n}} {{{\left| {u(x)} \right|}^p}dx} > 0, \quad {\alpha _\infty } > 0$ .

Obviously,

If this is the case, then either
$\displaystyle\mathop {\lim \inf }\limits_{m \to \infty } \mathop {\sup }\limits_{y \in {\mathbb{R}^n}} \int_{{B_R}(y)} {{{\left| {{u_m}(x)} \right|}^p}dx} = 0$

for all $R>0$ or

$\displaystyle\mathop {\lim \inf }\limits_{m \to \infty } \mathop {\sup }\limits_{y \in {\mathbb{R}^n}} \int_{{B_R}(y)} {{{\left| {{u_m}(x)} \right|}^p}dx} > 0$

for some $R>0$ . In the first case, by the Lions lemma [here], we deduce that

$\displaystyle {u_m} \to 0$

in $L^q(\mathbb R^n)$ for all $2<q<2^\star$ . In the second case, there exists a subsequence of $\{u_m\}$ relabelled again by $\{u_m\}$ , and a sequence $\{y_m\}\subset \mathbb R^n$ such that

$\displaystyle u_m(\cdot + y_m) \rightharpoonup u \not\equiv 0$

in $L^p(\mathbb R^n)$ .
If this is the case, then the sequence $\{|u_m|^p\}$ is tight.
If this is the case, we get the dichotomy property. The way to see this is to use the Lévy concentration functions $Q_m(t)$ defined by
$\displaystyle {Q_m}(t) = \mathop {\sup }\limits_{x \in {\mathbb{R}^n}} \int_{{B_t}(x)} {{{\left| {{u_m}(y)} \right|}^p}dy} , \quad m \geqslant 1, \quad t > 0$ .

Each function $Q_m$ is increasing on $[0,\infty)$ and by the Helly selection theorem

$Q(t)=\mathop {\lim }\limits_{m \to \infty} Q_m(t)$

is an increasing function on $[0,\infty)$ . Since $0<\alpha_\infty<\lambda$ , it is easy to check that

$0< \mathop {\lim }\limits_{t \to \infty} Q(t) < \lambda$ .

Remark. The dichotomy property tells us that sequence $\{u_m\}$ can be split into two parts concentrating on two disjoint sets whose distance tends to $\infty$ as $m\to \infty$ .

Remark. It is easy to find that if compactness happens then neither vanishing nor dichotomy can happen, but vanishing and dichotomy can happen together. In fact, it had been proved that up to a subsequence of $\{\rho_m\}_m$ either compactness or dichotomy happens. This result was due to Xing, Yi and Yaotian [here].

In practice, we look for a minimizing sequence of an energy functional. We then try to identify which of compactness, vanishing or dichotomy happens. Once this step is done, we can solve the problem (most of the time, we wish the dichotomy cannot occur). I will demonstrate several applications later.

Leave a Comment »

No comments yet.

RSS feed for comments on this post. TrackBack URI

M	T	W	T	F	S	S
« Apr				Jun »
					1	2
3	4	5	6	7	8	9
10	11	12	13	14	15	16
17	18	19	20	21	22	23
24	25	26	27	28	29	30
31

	Ngô Quốc Anh on The Poincaré inequality: W^{1,…
	Paul on The Poincaré inequality: W^{1,…
	Ngô Quốc Anh on A lower bound for solutions in…
	fanicolas on A lower bound for solutions in…
	Ngô Quốc Anh on The Alexandrov-Bol inequa…
	Fran on A lower bound for solutions in…
	Fab on The Alexandrov-Bol inequa…
	Trần Lượng on “Tích phân suy rộng…
	Trần Lượng on “Tích phân suy rộng…
	Ngô Quốc Anh on Contact me

Ngô Quốc Anh

May 28, 2010