Theorem in Adams Sobolev spaces book requires $uin L^p(Omega) cap L^r(Omega)$ but we only have $u in C^infty$...
$begingroup$
Let $Omega$ be a (open) domain in $mathbb{R}^n$. In Theorem 4.19 of Adams book on Sobolev spaces he makes use of Theorem 2.11 (An Interpolation Inequality). Theorem 2.11 requires that if we have $1le p < q < r$ and $uin L^p(Omega) cap L^r(Omega)$, then we have that $uin L^q(Omega)$ and
$$
||u||_q le ||u||_p^theta ||u||_r^{1-theta},
$$
for $0 < theta < 1$.
However in Theorem 4.19 we only have $uin C^infty(Omega)$ so how can he apply Theorem 2.11?
I don't know if it makes any difference, but he also states that $u$ and all its derivatives are extended by zero outside $Omega$ in Theorem 4.19. Is it this extension by zero that lets him have know that $uin L^p(Omega) cap L^r(Omega)$ and thus apply Theorem 2.11?
functional-analysis pde sobolev-spaces lp-spaces
asked Nov 29 '18 at 9:16
sonicboomsonicboom
3,68582753
$endgroup$
add a comment |
$begingroup$
Let $Omega$ be a (open) domain in $mathbb{R}^n$. In Theorem 4.19 of Adams book on Sobolev spaces he makes use of Theorem 2.11 (An Interpolation Inequality). Theorem 2.11 requires that if we have $1le p < q < r$ and $uin L^p(Omega) cap L^r(Omega)$, then we have that $uin L^q(Omega)$ and
$$
||u||_q le ||u||_p^theta ||u||_r^{1-theta},
$$
for $0 < theta < 1$.
However in Theorem 4.19 we only have $uin C^infty(Omega)$ so how can he apply Theorem 2.11?
I don't know if it makes any difference, but he also states that $u$ and all its derivatives are extended by zero outside $Omega$ in Theorem 4.19. Is it this extension by zero that lets him have know that $uin L^p(Omega) cap L^r(Omega)$ and thus apply Theorem 2.11?
functional-analysis pde sobolev-spaces lp-spaces
asked Nov 29 '18 at 9:16
sonicboomsonicboom
3,68582753
$endgroup$
1
$begingroup$
You might say that the inequality is obviously true (but useless) if $unotin L^p$ or $unotin L^q$ since the right-hand side is infinite. You can't conclude that $uin L^q$ of course. That with the extension outside of $Omega$ can't be an important fact since after all $Omega=mathbb{R}^n$ is allowed.
$endgroup$
– Olivier Moschetta
Nov 29 '18 at 9:20
$begingroup$
Actually I see how it all works out now from the rest of the theorem so nevermind.
$endgroup$
– sonicboom
Nov 29 '18 at 9:52
2
$begingroup$
@sonicboom then maybe you should post an answer, so that this question does not appear as unanswered
$endgroup$
– supinf
Nov 29 '18 at 10:31
add a comment |
$begingroup$
Let $Omega$ be a (open) domain in $mathbb{R}^n$. In Theorem 4.19 of Adams book on Sobolev spaces he makes use of Theorem 2.11 (An Interpolation Inequality). Theorem 2.11 requires that if we have $1le p < q < r$ and $uin L^p(Omega) cap L^r(Omega)$, then we have that $uin L^q(Omega)$ and
$$
||u||_q le ||u||_p^theta ||u||_r^{1-theta},
$$
for $0 < theta < 1$.
However in Theorem 4.19 we only have $uin C^infty(Omega)$ so how can he apply Theorem 2.11?
I don't know if it makes any difference, but he also states that $u$ and all its derivatives are extended by zero outside $Omega$ in Theorem 4.19. Is it this extension by zero that lets him have know that $uin L^p(Omega) cap L^r(Omega)$ and thus apply Theorem 2.11?
functional-analysis pde sobolev-spaces lp-spaces
asked Nov 29 '18 at 9:16
sonicboomsonicboom
3,68582753
$endgroup$
Let $Omega$ be a (open) domain in $mathbb{R}^n$. In Theorem 4.19 of Adams book on Sobolev spaces he makes use of Theorem 2.11 (An Interpolation Inequality). Theorem 2.11 requires that if we have $1le p < q < r$ and $uin L^p(Omega) cap L^r(Omega)$, then we have that $uin L^q(Omega)$ and
$$
||u||_q le ||u||_p^theta ||u||_r^{1-theta},
$$
for $0 < theta < 1$.
However in Theorem 4.19 we only have $uin C^infty(Omega)$ so how can he apply Theorem 2.11?
I don't know if it makes any difference, but he also states that $u$ and all its derivatives are extended by zero outside $Omega$ in Theorem 4.19. Is it this extension by zero that lets him have know that $uin L^p(Omega) cap L^r(Omega)$ and thus apply Theorem 2.11?
functional-analysis pde sobolev-spaces lp-spaces
functional-analysis pde sobolev-spaces lp-spaces
asked Nov 29 '18 at 9:16
sonicboomsonicboom
3,68582753
asked Nov 29 '18 at 9:16
sonicboomsonicboom
3,68582753
asked Nov 29 '18 at 9:16
sonicboomsonicboom
3,68582753
asked Nov 29 '18 at 9:16
sonicboomsonicboom
3,68582753
asked Nov 29 '18 at 9:16
sonicboomsonicboom
3,68582753
3,68582753
1
$begingroup$
You might say that the inequality is obviously true (but useless) if $unotin L^p$ or $unotin L^q$ since the right-hand side is infinite. You can't conclude that $uin L^q$ of course. That with the extension outside of $Omega$ can't be an important fact since after all $Omega=mathbb{R}^n$ is allowed.
$endgroup$
– Olivier Moschetta
Nov 29 '18 at 9:20
$begingroup$
Actually I see how it all works out now from the rest of the theorem so nevermind.
$endgroup$
– sonicboom
Nov 29 '18 at 9:52
2
$begingroup$
@sonicboom then maybe you should post an answer, so that this question does not appear as unanswered
$endgroup$
– supinf
Nov 29 '18 at 10:31
add a comment |
1
$begingroup$
You might say that the inequality is obviously true (but useless) if $unotin L^p$ or $unotin L^q$ since the right-hand side is infinite. You can't conclude that $uin L^q$ of course. That with the extension outside of $Omega$ can't be an important fact since after all $Omega=mathbb{R}^n$ is allowed.
$endgroup$
– Olivier Moschetta
Nov 29 '18 at 9:20
$begingroup$
Actually I see how it all works out now from the rest of the theorem so nevermind.
$endgroup$
– sonicboom
Nov 29 '18 at 9:52
2
$begingroup$
@sonicboom then maybe you should post an answer, so that this question does not appear as unanswered
$endgroup$
– supinf
Nov 29 '18 at 10:31
1
1
$begingroup$
You might say that the inequality is obviously true (but useless) if $unotin L^p$ or $unotin L^q$ since the right-hand side is infinite. You can't conclude that $uin L^q$ of course. That with the extension outside of $Omega$ can't be an important fact since after all $Omega=mathbb{R}^n$ is allowed.
$endgroup$
– Olivier Moschetta
Nov 29 '18 at 9:20
$begingroup$
You might say that the inequality is obviously true (but useless) if $unotin L^p$ or $unotin L^q$ since the right-hand side is infinite. You can't conclude that $uin L^q$ of course. That with the extension outside of $Omega$ can't be an important fact since after all $Omega=mathbb{R}^n$ is allowed.
$endgroup$
– Olivier Moschetta
Nov 29 '18 at 9:20
$begingroup$
Actually I see how it all works out now from the rest of the theorem so nevermind.
$endgroup$
– sonicboom
Nov 29 '18 at 9:52
$begingroup$
Actually I see how it all works out now from the rest of the theorem so nevermind.
$endgroup$
– sonicboom
Nov 29 '18 at 9:52
2
2
$begingroup$
@sonicboom then maybe you should post an answer, so that this question does not appear as unanswered
$endgroup$
– supinf
Nov 29 '18 at 10:31
$begingroup$
@sonicboom then maybe you should post an answer, so that this question does not appear as unanswered
$endgroup$
– supinf
Nov 29 '18 at 10:31
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
When Adams states that $u in C^infty(Omega)$ in the opening paragraph of this theorem he doing so to show that we have inequality (13).
He then estimates an $L^p$ norm of $u$ and can make use of inequality (13) through the density of the $C^infty$ functions in $L^p$ spaces.
answered Nov 30 '18 at 17:49
sonicboomsonicboom
3,68582753
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3018391%2ftheorem-in-adams-sobolev-spaces-book-requires-u-in-lp-omega-cap-lr-omega%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
When Adams states that $u in C^infty(Omega)$ in the opening paragraph of this theorem he doing so to show that we have inequality (13).
He then estimates an $L^p$ norm of $u$ and can make use of inequality (13) through the density of the $C^infty$ functions in $L^p$ spaces.
answered Nov 30 '18 at 17:49
sonicboomsonicboom
3,68582753
$endgroup$
add a comment |
$begingroup$
When Adams states that $u in C^infty(Omega)$ in the opening paragraph of this theorem he doing so to show that we have inequality (13).
He then estimates an $L^p$ norm of $u$ and can make use of inequality (13) through the density of the $C^infty$ functions in $L^p$ spaces.
answered Nov 30 '18 at 17:49
sonicboomsonicboom
3,68582753
$endgroup$
add a comment |
$begingroup$
When Adams states that $u in C^infty(Omega)$ in the opening paragraph of this theorem he doing so to show that we have inequality (13).
He then estimates an $L^p$ norm of $u$ and can make use of inequality (13) through the density of the $C^infty$ functions in $L^p$ spaces.
answered Nov 30 '18 at 17:49
sonicboomsonicboom
3,68582753
$endgroup$
When Adams states that $u in C^infty(Omega)$ in the opening paragraph of this theorem he doing so to show that we have inequality (13).
He then estimates an $L^p$ norm of $u$ and can make use of inequality (13) through the density of the $C^infty$ functions in $L^p$ spaces.
answered Nov 30 '18 at 17:49
sonicboomsonicboom
3,68582753
answered Nov 30 '18 at 17:49
sonicboomsonicboom
3,68582753
answered Nov 30 '18 at 17:49
sonicboomsonicboom
3,68582753
answered Nov 30 '18 at 17:49
sonicboomsonicboom
3,68582753
3,68582753
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3018391%2ftheorem-in-adams-sobolev-spaces-book-requires-u-in-lp-omega-cap-lr-omega%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
You might say that the inequality is obviously true (but useless) if $unotin L^p$ or $unotin L^q$ since the right-hand side is infinite. You can't conclude that $uin L^q$ of course. That with the extension outside of $Omega$ can't be an important fact since after all $Omega=mathbb{R}^n$ is allowed.
$endgroup$
– Olivier Moschetta
Nov 29 '18 at 9:20
$begingroup$
Actually I see how it all works out now from the rest of the theorem so nevermind.
$endgroup$
– sonicboom
Nov 29 '18 at 9:52
2
$begingroup$
@sonicboom then maybe you should post an answer, so that this question does not appear as unanswered
$endgroup$
– supinf
Nov 29 '18 at 10:31