Maximum Principle for Elliptic Problems
I am reading Renardy - "An Introduction to PDEs" where I am at the maximum principle for elliptic equations. The strong maximum principle states:
Assume $Luge0 (Lule0)$ in $Omega$ and assume that $u$ is non constant. If $c=0$ then $u$ does not achieve its maximum (minimum) in the interior of $Omega$. If $cle 0$, $u$ cannot achieve a non-negative maximum (non-positive minimum) in the interior. Regardless of the sign of $c$, $u$ cannot be zero at an interior maximum (minimum).
The operator $L$ is elliptic and $Omega$ is the domain. The exact assumptions on the domain and operator will not be given now, but are assumed to hold for my question.
I now want to try it on the equation
$begin{align}
Delta u(x,y)&=u(x,y)+sin(xy), &(x,y)in D_0(1)\
u(x,y)&=1, &(x,y)inpartial D_0(1)
end{align}$
where $D_0(1)$ is the unit disk centered around the origin.
I want to give an estimate of $sup|u|$ using the strong maximum principle (and possibly the weak one, too).
My attempt:
The requirements for the theorem are satisfied, as $Omega = D_0(1)$ and $L$ is elliptic. (But this shall not bother us now)
Since it is also required that $u$ is sufficiently smooth (has to be $mathcal{C}^2$ in the interior) the maximum and supremum coincide.
I obviously have to use the part of the theorem where $cle0$. But here already the first problem occurs, I do not have $Lule 0$ or $Luge 0$.
pde
asked Jan 4 at 12:49
EpsilonDeltaEpsilonDelta
6281615
add a comment |
I am reading Renardy - "An Introduction to PDEs" where I am at the maximum principle for elliptic equations. The strong maximum principle states:
Assume $Luge0 (Lule0)$ in $Omega$ and assume that $u$ is non constant. If $c=0$ then $u$ does not achieve its maximum (minimum) in the interior of $Omega$. If $cle 0$, $u$ cannot achieve a non-negative maximum (non-positive minimum) in the interior. Regardless of the sign of $c$, $u$ cannot be zero at an interior maximum (minimum).
The operator $L$ is elliptic and $Omega$ is the domain. The exact assumptions on the domain and operator will not be given now, but are assumed to hold for my question.
I now want to try it on the equation
$begin{align}
Delta u(x,y)&=u(x,y)+sin(xy), &(x,y)in D_0(1)\
u(x,y)&=1, &(x,y)inpartial D_0(1)
end{align}$
where $D_0(1)$ is the unit disk centered around the origin.
I want to give an estimate of $sup|u|$ using the strong maximum principle (and possibly the weak one, too).
My attempt:
The requirements for the theorem are satisfied, as $Omega = D_0(1)$ and $L$ is elliptic. (But this shall not bother us now)
Since it is also required that $u$ is sufficiently smooth (has to be $mathcal{C}^2$ in the interior) the maximum and supremum coincide.
I obviously have to use the part of the theorem where $cle0$. But here already the first problem occurs, I do not have $Lule 0$ or $Luge 0$.
pde
asked Jan 4 at 12:49
EpsilonDeltaEpsilonDelta
6281615
add a comment |
I am reading Renardy - "An Introduction to PDEs" where I am at the maximum principle for elliptic equations. The strong maximum principle states:
Assume $Luge0 (Lule0)$ in $Omega$ and assume that $u$ is non constant. If $c=0$ then $u$ does not achieve its maximum (minimum) in the interior of $Omega$. If $cle 0$, $u$ cannot achieve a non-negative maximum (non-positive minimum) in the interior. Regardless of the sign of $c$, $u$ cannot be zero at an interior maximum (minimum).
The operator $L$ is elliptic and $Omega$ is the domain. The exact assumptions on the domain and operator will not be given now, but are assumed to hold for my question.
I now want to try it on the equation
$begin{align}
Delta u(x,y)&=u(x,y)+sin(xy), &(x,y)in D_0(1)\
u(x,y)&=1, &(x,y)inpartial D_0(1)
end{align}$
where $D_0(1)$ is the unit disk centered around the origin.
I want to give an estimate of $sup|u|$ using the strong maximum principle (and possibly the weak one, too).
My attempt:
The requirements for the theorem are satisfied, as $Omega = D_0(1)$ and $L$ is elliptic. (But this shall not bother us now)
Since it is also required that $u$ is sufficiently smooth (has to be $mathcal{C}^2$ in the interior) the maximum and supremum coincide.
I obviously have to use the part of the theorem where $cle0$. But here already the first problem occurs, I do not have $Lule 0$ or $Luge 0$.
pde
asked Jan 4 at 12:49
EpsilonDeltaEpsilonDelta
6281615
I am reading Renardy - "An Introduction to PDEs" where I am at the maximum principle for elliptic equations. The strong maximum principle states:
Assume $Luge0 (Lule0)$ in $Omega$ and assume that $u$ is non constant. If $c=0$ then $u$ does not achieve its maximum (minimum) in the interior of $Omega$. If $cle 0$, $u$ cannot achieve a non-negative maximum (non-positive minimum) in the interior. Regardless of the sign of $c$, $u$ cannot be zero at an interior maximum (minimum).
The operator $L$ is elliptic and $Omega$ is the domain. The exact assumptions on the domain and operator will not be given now, but are assumed to hold for my question.
I now want to try it on the equation
$begin{align}
Delta u(x,y)&=u(x,y)+sin(xy), &(x,y)in D_0(1)\
u(x,y)&=1, &(x,y)inpartial D_0(1)
end{align}$
where $D_0(1)$ is the unit disk centered around the origin.
I want to give an estimate of $sup|u|$ using the strong maximum principle (and possibly the weak one, too).
My attempt:
The requirements for the theorem are satisfied, as $Omega = D_0(1)$ and $L$ is elliptic. (But this shall not bother us now)
Since it is also required that $u$ is sufficiently smooth (has to be $mathcal{C}^2$ in the interior) the maximum and supremum coincide.
I obviously have to use the part of the theorem where $cle0$. But here already the first problem occurs, I do not have $Lule 0$ or $Luge 0$.
pde
pde
asked Jan 4 at 12:49
EpsilonDeltaEpsilonDelta
6281615
asked Jan 4 at 12:49
EpsilonDeltaEpsilonDelta
6281615
asked Jan 4 at 12:49
EpsilonDeltaEpsilonDelta
6281615
asked Jan 4 at 12:49
EpsilonDeltaEpsilonDelta
6281615
asked Jan 4 at 12:49
EpsilonDeltaEpsilonDelta
6281615
6281615
add a comment |
add a comment |
0
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',
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%2f3061625%2fmaximum-principle-for-elliptic-problems%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.
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%2f3061625%2fmaximum-principle-for-elliptic-problems%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
