# 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.