Regularity for weak solution of Poisson problem in a rectangle
$begingroup$
Let $Omega=(0,1)^2$. Let $u$ be a weak solution of $Delta u=f$ con $f in L^2(Omega)$ e $u in H^1_0(Omega)$. I would like to prove that $u in H^2(Omega)$.
I know that $u in H^2_{loc}(Omega)$ because of elliptic interior regularity. I tried to adapt the demonstration of that fact and to find compact subsets $K_n$ with $|u|_{H^2(K_n)}$ uniformly bounded, but I was not able to go any further.
As a reference to understand what I know about the subject(which is very few) I've studied the chapter about that of Evans book and of Brezis book.
sobolev-spaces harmonic-functions elliptic-equations
asked Jan 12 at 16:31
Tommaso ScognamiglioTommaso Scognamiglio
591412
$endgroup$
add a comment |
$begingroup$
Let $Omega=(0,1)^2$. Let $u$ be a weak solution of $Delta u=f$ con $f in L^2(Omega)$ e $u in H^1_0(Omega)$. I would like to prove that $u in H^2(Omega)$.
I know that $u in H^2_{loc}(Omega)$ because of elliptic interior regularity. I tried to adapt the demonstration of that fact and to find compact subsets $K_n$ with $|u|_{H^2(K_n)}$ uniformly bounded, but I was not able to go any further.
As a reference to understand what I know about the subject(which is very few) I've studied the chapter about that of Evans book and of Brezis book.
sobolev-spaces harmonic-functions elliptic-equations
asked Jan 12 at 16:31
Tommaso ScognamiglioTommaso Scognamiglio
591412
$endgroup$
$begingroup$
Concerning this subject, I would recommend Grisvard's "Elliptic Problems in Nonsmooth Domains". If you have access, you can download it at doi.org/10.1137/1.9781611972030
$endgroup$
– gerw
Jan 14 at 11:01
$begingroup$
I don't have access but I'll try to take a look at the book .Thanks
$endgroup$
– Tommaso Scognamiglio
Jan 14 at 15:42
add a comment |
$begingroup$
Let $Omega=(0,1)^2$. Let $u$ be a weak solution of $Delta u=f$ con $f in L^2(Omega)$ e $u in H^1_0(Omega)$. I would like to prove that $u in H^2(Omega)$.
I know that $u in H^2_{loc}(Omega)$ because of elliptic interior regularity. I tried to adapt the demonstration of that fact and to find compact subsets $K_n$ with $|u|_{H^2(K_n)}$ uniformly bounded, but I was not able to go any further.
As a reference to understand what I know about the subject(which is very few) I've studied the chapter about that of Evans book and of Brezis book.
sobolev-spaces harmonic-functions elliptic-equations
asked Jan 12 at 16:31
Tommaso ScognamiglioTommaso Scognamiglio
591412
$endgroup$
Let $Omega=(0,1)^2$. Let $u$ be a weak solution of $Delta u=f$ con $f in L^2(Omega)$ e $u in H^1_0(Omega)$. I would like to prove that $u in H^2(Omega)$.
I know that $u in H^2_{loc}(Omega)$ because of elliptic interior regularity. I tried to adapt the demonstration of that fact and to find compact subsets $K_n$ with $|u|_{H^2(K_n)}$ uniformly bounded, but I was not able to go any further.
As a reference to understand what I know about the subject(which is very few) I've studied the chapter about that of Evans book and of Brezis book.
sobolev-spaces harmonic-functions elliptic-equations
sobolev-spaces harmonic-functions elliptic-equations
asked Jan 12 at 16:31
Tommaso ScognamiglioTommaso Scognamiglio
591412
asked Jan 12 at 16:31
Tommaso ScognamiglioTommaso Scognamiglio
591412
asked Jan 12 at 16:31
Tommaso ScognamiglioTommaso Scognamiglio
591412
asked Jan 12 at 16:31
Tommaso ScognamiglioTommaso Scognamiglio
591412
asked Jan 12 at 16:31
Tommaso ScognamiglioTommaso Scognamiglio
591412
591412
$begingroup$
Concerning this subject, I would recommend Grisvard's "Elliptic Problems in Nonsmooth Domains". If you have access, you can download it at doi.org/10.1137/1.9781611972030
$endgroup$
– gerw
Jan 14 at 11:01
$begingroup$
I don't have access but I'll try to take a look at the book .Thanks
$endgroup$
– Tommaso Scognamiglio
Jan 14 at 15:42
add a comment |
$begingroup$
Concerning this subject, I would recommend Grisvard's "Elliptic Problems in Nonsmooth Domains". If you have access, you can download it at doi.org/10.1137/1.9781611972030
$endgroup$
– gerw
Jan 14 at 11:01
$begingroup$
I don't have access but I'll try to take a look at the book .Thanks
$endgroup$
– Tommaso Scognamiglio
Jan 14 at 15:42
$begingroup$
Concerning this subject, I would recommend Grisvard's "Elliptic Problems in Nonsmooth Domains". If you have access, you can download it at doi.org/10.1137/1.9781611972030
$endgroup$
– gerw
Jan 14 at 11:01
$begingroup$
Concerning this subject, I would recommend Grisvard's "Elliptic Problems in Nonsmooth Domains". If you have access, you can download it at doi.org/10.1137/1.9781611972030
$endgroup$
– gerw
Jan 14 at 11:01
$begingroup$
I don't have access but I'll try to take a look at the book .Thanks
$endgroup$
– Tommaso Scognamiglio
Jan 14 at 15:42
$begingroup$
I don't have access but I'll try to take a look at the book .Thanks
$endgroup$
– Tommaso Scognamiglio
Jan 14 at 15:42
add a comment |
0
active
oldest
votes
Your Answer
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%2f3071087%2fregularity-for-weak-solution-of-poisson-problem-in-a-rectangle%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
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.
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%2f3071087%2fregularity-for-weak-solution-of-poisson-problem-in-a-rectangle%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
$begingroup$
Concerning this subject, I would recommend Grisvard's "Elliptic Problems in Nonsmooth Domains". If you have access, you can download it at doi.org/10.1137/1.9781611972030
$endgroup$
– gerw
Jan 14 at 11:01
$begingroup$
I don't have access but I'll try to take a look at the book .Thanks
$endgroup$
– Tommaso Scognamiglio
Jan 14 at 15:42