Understanding the proof of the Concentration-Compactness principle

Question

$\newcommand{\R}{\mathbb{R}}$ I'm reading parts of the paper The Concentration-Compactness Principle in the Calculus of Variations. The Limit Case, Part 1 by P.L. Lions. I'm trying to understand the proof of Lemma 1.1, but I'm having some trouble. The hypotheses of the lemma are the following:

Let $(u_n)_n$ be a bounded sequence in $W^{m,p}_0(\R^N)$ converging weakly to some $u$ and such that $|D^m u_n|^p$ converges weakly to $\mu$, and $|u_n|^p$ converges tightly to $\nu$, where $\mu,\nu$ are bounded nonnegative measures on $\R^N$.

The proof (p.160, or p.16 in the PDF) starts off by letting $\phi \in C^{\infty}_c(\R^N)$ and applying Sobolev's inequality to $\phi u_n$:

\begin{align*} \left( \int \limits_{\R^N} |\phi|^q |u_n|^q dx \right)^{1/q} \leq C \left( \int \limits_{\R^N} |D^m(\phi u_n)|^p dx \right)^{1/p} \end{align*}

Then, there is the following argument:

The right-hand side is estimated as follows:

$$ \left| \left( \int_{\R^N} |D^m(\phi u_n)|^p dx \right)^{1/p} - \left( \int_{\R^N} |\phi|^p |D^m u_n|^p \right)^{1/p} \right| \\ \leq C \sum \limits_{j=0}^{m-1} \left( \int_{\R^N} |D^{m-j}\phi|^p |D^j u_n|^p dx \right)^{1/p} $$ And using the fact that $\phi$ has compact support and the Rellich theorem, we see that this bound goes to $0$ as $n$ goes to infinity.

Questions:

1. Where does the estimate come from? I tried to expand the derivative of the product $\phi u_n$, but it didn't get me anywhere.

2. How do we conclude that bound goes to $0$? I know the Rellich theorem is about compact embeddings of $W^{1,p}$ into $L^r$ for appropriate values of $r$, but I don't see how it helps. Is there a version for the general space $W^{m,p}$?

A final point: the paper defines $|D^m \phi(x)|$ as "any product norm of all derivatives of order $m$ at the point $x$". Is this different from the definition in Evans' book, i.e. $|D^m \phi| = (\sum_{|\alpha|=m} |D^{\alpha} \phi|^2)^{1/2}$?

Answer 1

1. Expanding was the correct idea. Note that we have by the product rule $$D^m (\phi un) = \sum_{j=0}^m \binom{m}{j} D^{m-j} \phi \, D^j u_n,$$ thus we can cancel the $m$-th term in the sum using the reverse triangle inequality $$\begin{aligned} \big| \|D^m(\phi u_n) \|_{L^p} - \|\phi D^m u_n \|_{L^p} \big|\leq \|D^m(\phi u_n)-\phi D^m u_n\|_{L^p} \leq C\sum_{j=0}^{m-1} \|D^{m-j} \phi \, D^j u_n\|_{L^p}.\end{aligned}$$

2. Yes, there is such a version of Rellich's theorem. Indeed, $W_0^{m,p}$ is compactly embedded in $W_0^{m-1,p}$ for a bounded domain. Now, since we cancelled the $m$-th sum in the estimate, we can directly apply Rellich. See the book "Linear Functional Analysis" by H.W. Alt.

3. Indeed, it is defined in the same way.