Ngô Quốc Anh

September 28, 2011

Concentration-Compactness principle, II

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 9:35

In this entry, we continue to talk about the Concentration-Compactness Principle discovered by P.L. Lions [here]. In the previous entry, we already discussed two forms of non-compactness due to unbounded domains. Here we discuss what happens when passing to the limit on those functionals along weakly convergent subsequences.

Theorem (Lions). Let \{u_j\}_j be sequence in D^{1,p}(\mathbb R^n) weakly convergent to u and such that

  • \{|\nabla u_j\|^p\} converges weak* to a nonnegative measure \mu,
  • \{|u_j|^{p^\star}\} converges weak* to a nonnegative measure \nu.

Then there exists an at most countable index set J, sequence \{x_j\} \subset \mathbb R^n, \{\mu_j\}, \{\nu_j\} \subset (0,\infty), j \in J, such that

\displaystyle\nu = |u{|^{{p^ \star }}} + \sum\limits_{j \in J} {{\nu _j}{\delta _{{x_j}}}} ,

and

\displaystyle\mu \geqslant |\nabla u{|^p} + \sum\limits_{j \in J} {{\mu _j}{\delta _{{x_j}}}} ,

and

\displaystyle S\nu _j^{\frac{p}{{{p^ \star }}}} \leqslant {\mu _j},

where S is the best Sobolev constant and \delta_{x_j} are Dirac measures assigned to x_j. If u \equiv 0 and

\displaystyle \int_{{\mathbb{R}^n}} {d\mu } \leqslant S{\left( {\int_{{\mathbb{R}^n}} {d\nu } } \right)^{\frac{p}{{{p^ \star }}}}}

then J is a singleton and

\displaystyle\nu=\gamma\delta_{x_0}=\frac{1}{S}\gamma^\frac{p}{n}\mu

for some \gamma \geqslant 0.

Apparently, the theorem does not provide any information about possible loss of mass at infinity of a weakly convergent minimizing sequence. We shall consider that case in the forthcoming topic.

See also:

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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.

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May 13, 2010

Concentration-Compactness Principle: The loss of mass at infinity in the subcritical case

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 16:18

One of the main problems encountered in solving variational problems is to show the convergence of minimizing sequence (the direct method). In many problems, it is relatively easy to show the boundedness of a minimizing sequence in an appropriate Sobolev space.

Th aim of this section is to derive the concentration-compactness at infinity (just for subcritical case). Roughly speaking, for unbounded domains due to the lack of compact embedding in the Sobolev embedding theorem, the loss of mass of a weakly convergent sequence may occur. In fact, this can only occur at infinity.

Let us firstly consider the case \Omega \subset \mathbb R^n is a bounded domain. Let \{u_m\} be a sequence in W_0^{1,2}(\Omega) such that

u_m \rightharpoonup u

weakly in W_0^{1,2}(\Omega) (i.e. u \in W_0^{1,2}(\Omega) and f(u_m) \to f(u) for any f \in W_0^{-1,2}(\Omega)). Assume 2<p<2^\star. The Sobolev embedding tells us that

W_0^{1,2}(\Omega) \hookrightarrow L^p(\Omega)

is compact. Thus

u_m \to u

strongly in L^p(\Omega) in the sense that

\displaystyle\mathop {\lim }\limits_{m \to \infty } \int_\Omega {{{\left| {{u_m} - u} \right|}^p}dx} = 0.

It now follows from the Brezis-Lieb lemma or the use of the Minkowski inequality

\displaystyle {\left( {\int_\Omega {{{\left| {{u_m}} \right|}^p}dx} } \right)^{\frac{1}{p}}} \leqslant {\left( {\int_\Omega {{{\left| {{u_m} - u} \right|}^p}dx} } \right)^{\frac{1}{p}}} + {\left( {\int_\Omega {{{\left| u \right|}^p}dx} } \right)^{\frac{1}{p}}}

and

\displaystyle {\left( {\int_\Omega {{{\left| u \right|}^p}dx} } \right)^{\frac{1}{p}}} \leqslant {\left( {\int_\Omega {{{\left| {u - {u_m}} \right|}^p}dx} } \right)^{\frac{1}{p}}} + {\left( {\int_\Omega {{{\left| {{u_m}} \right|}^p}dx} } \right)^{\frac{1}{p}}}

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