Ngô Quốc Anh

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.

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

  1. \alpha _\infty=\lambda and thus u \equiv 0 (almost everywhere)
  2. \alpha _\infty=0 and thus

    \displaystyle\lambda = \int_{{\mathbb{R}^n}} {{{\left| {u(x)} \right|}^p}dx}

  3. and

    \displaystyle\int_{{\mathbb{R}^n}} {{{\left| {u(x)} \right|}^p}dx} > 0, \quad {\alpha _\infty } > 0.

Obviously,

  1. 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).

  2. If this is the case, then the sequence \{|u_m|^p\} is tight.
  3. 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.

See also:

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