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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