Mixing
Dirichlet Problem with additional Neumann boundary condition
$begingroup$
I am dealing with Dirichlet problems - i.e. minimizing the energy functional, but with Neumann boundary condition - i.e. zero derivative - on the whole boundary and additionally with Dirichlet boundary condition - i.e. fixed value - on part of the boundary.
The difference to usual mixed boundary conditions (e.g. Zaremba's problem) is that the boundary conditions
do not scope separate parts of the boundary but overlap.
I vaguely remember that I once stumbled over a paper dealing with this type of problem, but now I cannot find it any more.
This question is for literature hints regarding such problems.
I am not necessarily interested in a solution method, but mainly on properties of the solutions, e.g. are the solutions harmonic functions? Under what conditions does a solution exist etc. Actually I am studying an algorithm that produces results that can be interpreted as solutions of such problems. So it is relevant to me what regularity conditions this implies for the results.
differential-geometry reference-request pde
edited Feb 2 at 21:29
DisintegratingByParts
60.4k42681
asked Feb 2 at 21:18
steworistewori
1115
$endgroup$
add a comment |
$begingroup$
I am dealing with Dirichlet problems - i.e. minimizing the energy functional, but with Neumann boundary condition - i.e. zero derivative - on the whole boundary and additionally with Dirichlet boundary condition - i.e. fixed value - on part of the boundary.
The difference to usual mixed boundary conditions (e.g. Zaremba's problem) is that the boundary conditions
do not scope separate parts of the boundary but overlap.
I vaguely remember that I once stumbled over a paper dealing with this type of problem, but now I cannot find it any more.
This question is for literature hints regarding such problems.
I am not necessarily interested in a solution method, but mainly on properties of the solutions, e.g. are the solutions harmonic functions? Under what conditions does a solution exist etc. Actually I am studying an algorithm that produces results that can be interpreted as solutions of such problems. So it is relevant to me what regularity conditions this implies for the results.
differential-geometry reference-request pde
edited Feb 2 at 21:29
DisintegratingByParts
60.4k42681
asked Feb 2 at 21:18
steworistewori
1115
$endgroup$
$begingroup$
DisintegratingByParts, Thanks for your edit regarding Neumann vs von Neumann. That seems to be a common mistake: I frequently saw it as "von Neumann boundary condition" in other questions but also even in some papers. Will correct it in my own texts too...
$endgroup$
– stewori
Feb 4 at 10:31
add a comment |
$begingroup$
I am dealing with Dirichlet problems - i.e. minimizing the energy functional, but with Neumann boundary condition - i.e. zero derivative - on the whole boundary and additionally with Dirichlet boundary condition - i.e. fixed value - on part of the boundary.
The difference to usual mixed boundary conditions (e.g. Zaremba's problem) is that the boundary conditions
do not scope separate parts of the boundary but overlap.
I vaguely remember that I once stumbled over a paper dealing with this type of problem, but now I cannot find it any more.
This question is for literature hints regarding such problems.
I am not necessarily interested in a solution method, but mainly on properties of the solutions, e.g. are the solutions harmonic functions? Under what conditions does a solution exist etc. Actually I am studying an algorithm that produces results that can be interpreted as solutions of such problems. So it is relevant to me what regularity conditions this implies for the results.
differential-geometry reference-request pde
edited Feb 2 at 21:29
DisintegratingByParts
60.4k42681
asked Feb 2 at 21:18
steworistewori
1115
$endgroup$
I am dealing with Dirichlet problems - i.e. minimizing the energy functional, but with Neumann boundary condition - i.e. zero derivative - on the whole boundary and additionally with Dirichlet boundary condition - i.e. fixed value - on part of the boundary.
The difference to usual mixed boundary conditions (e.g. Zaremba's problem) is that the boundary conditions
do not scope separate parts of the boundary but overlap.
I vaguely remember that I once stumbled over a paper dealing with this type of problem, but now I cannot find it any more.
This question is for literature hints regarding such problems.
I am not necessarily interested in a solution method, but mainly on properties of the solutions, e.g. are the solutions harmonic functions? Under what conditions does a solution exist etc. Actually I am studying an algorithm that produces results that can be interpreted as solutions of such problems. So it is relevant to me what regularity conditions this implies for the results.
differential-geometry reference-request pde
differential-geometry reference-request pde
edited Feb 2 at 21:29
DisintegratingByParts
60.4k42681
asked Feb 2 at 21:18
steworistewori
1115
edited Feb 2 at 21:29
DisintegratingByParts
60.4k42681
asked Feb 2 at 21:18
steworistewori
1115
edited Feb 2 at 21:29
DisintegratingByParts
60.4k42681
edited Feb 2 at 21:29
DisintegratingByParts
60.4k42681
edited Feb 2 at 21:29
DisintegratingByParts
60.4k42681
60.4k42681
asked Feb 2 at 21:18
steworistewori
1115
asked Feb 2 at 21:18
steworistewori
1115
asked Feb 2 at 21:18
steworistewori
1115
1115
$begingroup$
DisintegratingByParts, Thanks for your edit regarding Neumann vs von Neumann. That seems to be a common mistake: I frequently saw it as "von Neumann boundary condition" in other questions but also even in some papers. Will correct it in my own texts too...
$endgroup$
– stewori
Feb 4 at 10:31
add a comment |
$begingroup$
DisintegratingByParts, Thanks for your edit regarding Neumann vs von Neumann. That seems to be a common mistake: I frequently saw it as "von Neumann boundary condition" in other questions but also even in some papers. Will correct it in my own texts too...
$endgroup$
– stewori
Feb 4 at 10:31
$begingroup$
DisintegratingByParts, Thanks for your edit regarding Neumann vs von Neumann. That seems to be a common mistake: I frequently saw it as "von Neumann boundary condition" in other questions but also even in some papers. Will correct it in my own texts too...
$endgroup$
– stewori
Feb 4 at 10:31
$begingroup$
DisintegratingByParts, Thanks for your edit regarding Neumann vs von Neumann. That seems to be a common mistake: I frequently saw it as "von Neumann boundary condition" in other questions but also even in some papers. Will correct it in my own texts too...
$endgroup$
– stewori
Feb 4 at 10:31
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%2f3097810%2fdirichlet-problem-with-additional-neumann-boundary-condition%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%2f3097810%2fdirichlet-problem-with-additional-neumann-boundary-condition%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
$begingroup$
DisintegratingByParts, Thanks for your edit regarding Neumann vs von Neumann. That seems to be a common mistake: I frequently saw it as "von Neumann boundary condition" in other questions but also even in some papers. Will correct it in my own texts too...
$endgroup$
– stewori
Feb 4 at 10:31