$H^1(Omega')$ is in genral not a subspace of $H^1 (Omega)$ for bounded domains $Omega' subset Omega$
up vote
0
down vote
favorite
Reading about Sobolev spaces I found the following statement:
$H^1 ({Omega}')$ is not a subspace of $H^1(Omega)$ for $Omega'subset Omega$.
$left(text{However } H^1_0 (Omega ')subset H^1_0(Omega)right)$
I guess that the reason behind is that you can not control $fin H^1(Omega ')$ outside $Omega'$. But I'm having a hard time thinking a counterexample or a more rigorous explanation.
functional-analysis pde sobolev-spaces
add a comment |
up vote
0
down vote
favorite
Reading about Sobolev spaces I found the following statement:
$H^1 ({Omega}')$ is not a subspace of $H^1(Omega)$ for $Omega'subset Omega$.
$left(text{However } H^1_0 (Omega ')subset H^1_0(Omega)right)$
I guess that the reason behind is that you can not control $fin H^1(Omega ')$ outside $Omega'$. But I'm having a hard time thinking a counterexample or a more rigorous explanation.
functional-analysis pde sobolev-spaces
1
can't you take something that blows up at the boundary?
– mathworker21
Nov 22 at 12:44
First of all, a function on $Omega'$ is not a function on $Omega$, so one has to say something to make sense of the inclusion in the first place. This is also the case (although easy) for $H^1_0 (Omega ')subset H^1_0(Omega)$. Please say something.
– Michał Miśkiewicz
Nov 22 at 15:36
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Reading about Sobolev spaces I found the following statement:
$H^1 ({Omega}')$ is not a subspace of $H^1(Omega)$ for $Omega'subset Omega$.
$left(text{However } H^1_0 (Omega ')subset H^1_0(Omega)right)$
I guess that the reason behind is that you can not control $fin H^1(Omega ')$ outside $Omega'$. But I'm having a hard time thinking a counterexample or a more rigorous explanation.
functional-analysis pde sobolev-spaces
Reading about Sobolev spaces I found the following statement:
$H^1 ({Omega}')$ is not a subspace of $H^1(Omega)$ for $Omega'subset Omega$.
$left(text{However } H^1_0 (Omega ')subset H^1_0(Omega)right)$
I guess that the reason behind is that you can not control $fin H^1(Omega ')$ outside $Omega'$. But I'm having a hard time thinking a counterexample or a more rigorous explanation.
functional-analysis pde sobolev-spaces
functional-analysis pde sobolev-spaces
edited Nov 22 at 12:27
gerw
18.9k11133
18.9k11133
asked Nov 22 at 12:24
gmirsan
63
63
1
can't you take something that blows up at the boundary?
– mathworker21
Nov 22 at 12:44
First of all, a function on $Omega'$ is not a function on $Omega$, so one has to say something to make sense of the inclusion in the first place. This is also the case (although easy) for $H^1_0 (Omega ')subset H^1_0(Omega)$. Please say something.
– Michał Miśkiewicz
Nov 22 at 15:36
add a comment |
1
can't you take something that blows up at the boundary?
– mathworker21
Nov 22 at 12:44
First of all, a function on $Omega'$ is not a function on $Omega$, so one has to say something to make sense of the inclusion in the first place. This is also the case (although easy) for $H^1_0 (Omega ')subset H^1_0(Omega)$. Please say something.
– Michał Miśkiewicz
Nov 22 at 15:36
1
1
can't you take something that blows up at the boundary?
– mathworker21
Nov 22 at 12:44
can't you take something that blows up at the boundary?
– mathworker21
Nov 22 at 12:44
First of all, a function on $Omega'$ is not a function on $Omega$, so one has to say something to make sense of the inclusion in the first place. This is also the case (although easy) for $H^1_0 (Omega ')subset H^1_0(Omega)$. Please say something.
– Michał Miśkiewicz
Nov 22 at 15:36
First of all, a function on $Omega'$ is not a function on $Omega$, so one has to say something to make sense of the inclusion in the first place. This is also the case (although easy) for $H^1_0 (Omega ')subset H^1_0(Omega)$. Please say something.
– Michał Miśkiewicz
Nov 22 at 15:36
add a comment |
active
oldest
votes
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',
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%2f3009067%2fh1-omega-is-in-genral-not-a-subspace-of-h1-omega-for-bounded-domains%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3009067%2fh1-omega-is-in-genral-not-a-subspace-of-h1-omega-for-bounded-domains%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
can't you take something that blows up at the boundary?
– mathworker21
Nov 22 at 12:44
First of all, a function on $Omega'$ is not a function on $Omega$, so one has to say something to make sense of the inclusion in the first place. This is also the case (although easy) for $H^1_0 (Omega ')subset H^1_0(Omega)$. Please say something.
– Michał Miśkiewicz
Nov 22 at 15:36