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Understanding the proof of the Concentration-Compactness principle
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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 $phi u_n$:
beginalign*
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
endalign*
Then, there is the following argument:
The right-hand side is estimated as follows:
$$
left|
left(
int_R^N |D^m(phi u_n)|^p dx
right)^1/p
-
left(
int_R^N |phi|^p |D^m u_n|^p
right)^1/p
right|
\
leq
C sum limits_j=0^m-1
left(
int_R^N |D^m-jphi|^p |D^j u_n|^p dx
right)^1/p
$$
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:
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.
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. $|D^m phi| = (sum_alpha |D^alpha phi|^2)^1/2$?
proof-explanation partial-derivative sobolev-spaces weak-convergence
asked Jul 20 at 22:18
Sambo
1,2561427
add a comment |Â
up vote
0
down vote
favorite
$newcommandRmathbbR$
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 $phi u_n$:
beginalign*
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
endalign*
Then, there is the following argument:
The right-hand side is estimated as follows:
$$
left|
left(
int_R^N |D^m(phi u_n)|^p dx
right)^1/p
-
left(
int_R^N |phi|^p |D^m u_n|^p
right)^1/p
right|
\
leq
C sum limits_j=0^m-1
left(
int_R^N |D^m-jphi|^p |D^j u_n|^p dx
right)^1/p
$$
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:
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.
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. $|D^m phi| = (sum_alpha |D^alpha phi|^2)^1/2$?
proof-explanation partial-derivative sobolev-spaces weak-convergence
asked Jul 20 at 22:18
Sambo
1,2561427
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
$newcommandRmathbbR$
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 $phi u_n$:
beginalign*
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
endalign*
Then, there is the following argument:
The right-hand side is estimated as follows:
$$
left|
left(
int_R^N |D^m(phi u_n)|^p dx
right)^1/p
-
left(
int_R^N |phi|^p |D^m u_n|^p
right)^1/p
right|
\
leq
C sum limits_j=0^m-1
left(
int_R^N |D^m-jphi|^p |D^j u_n|^p dx
right)^1/p
$$
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:
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.
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. $|D^m phi| = (sum_alpha |D^alpha phi|^2)^1/2$?
proof-explanation partial-derivative sobolev-spaces weak-convergence
asked Jul 20 at 22:18
Sambo
1,2561427
$newcommandRmathbbR$
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 $phi u_n$:
beginalign*
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
endalign*
Then, there is the following argument:
The right-hand side is estimated as follows:
$$
left|
left(
int_R^N |D^m(phi u_n)|^p dx
right)^1/p
-
left(
int_R^N |phi|^p |D^m u_n|^p
right)^1/p
right|
\
leq
C sum limits_j=0^m-1
left(
int_R^N |D^m-jphi|^p |D^j u_n|^p dx
right)^1/p
$$
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:
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.
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. $|D^m phi| = (sum_alpha |D^alpha phi|^2)^1/2$?
proof-explanation partial-derivative sobolev-spaces weak-convergence
asked Jul 20 at 22:18
Sambo
1,2561427
asked Jul 20 at 22:18
Sambo
1,2561427
asked Jul 20 at 22:18
Sambo
1,2561427
asked Jul 20 at 22:18
Sambo
1,2561427
1,2561427
add a comment |Â
add a comment |Â
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