Mixing
Why the definition of harmonic function needs twice “ continuously ” differentiable?
$begingroup$
Let $u$ denote a twice continuously differentiable function , then $u$ is harmonic if it satisfies Laplace's equation : $$Delta u=sum_{j=1}^d frac{partial^2u}{partial x_j^2}=0$$
However , if $f$ is a twice differentiable function which is not twice continuous differentiable , can $f$ satisfies that Laplace's equation ?
real-analysis harmonic-analysis
asked Feb 2 at 23:10
J.GuoJ.Guo
4579
$endgroup$
add a comment |
$begingroup$
Let $u$ denote a twice continuously differentiable function , then $u$ is harmonic if it satisfies Laplace's equation : $$Delta u=sum_{j=1}^d frac{partial^2u}{partial x_j^2}=0$$
However , if $f$ is a twice differentiable function which is not twice continuous differentiable , can $f$ satisfies that Laplace's equation ?
real-analysis harmonic-analysis
asked Feb 2 at 23:10
J.GuoJ.Guo
4579
$endgroup$
$begingroup$
See en.wikipedia.org/wiki/Harmonic_function
$endgroup$
– Kavi Rama Murthy
Feb 2 at 23:13
add a comment |
$begingroup$
Let $u$ denote a twice continuously differentiable function , then $u$ is harmonic if it satisfies Laplace's equation : $$Delta u=sum_{j=1}^d frac{partial^2u}{partial x_j^2}=0$$
However , if $f$ is a twice differentiable function which is not twice continuous differentiable , can $f$ satisfies that Laplace's equation ?
real-analysis harmonic-analysis
asked Feb 2 at 23:10
J.GuoJ.Guo
4579
$endgroup$
Let $u$ denote a twice continuously differentiable function , then $u$ is harmonic if it satisfies Laplace's equation : $$Delta u=sum_{j=1}^d frac{partial^2u}{partial x_j^2}=0$$
However , if $f$ is a twice differentiable function which is not twice continuous differentiable , can $f$ satisfies that Laplace's equation ?
real-analysis harmonic-analysis
real-analysis harmonic-analysis
asked Feb 2 at 23:10
J.GuoJ.Guo
4579
asked Feb 2 at 23:10
J.GuoJ.Guo
4579
asked Feb 2 at 23:10
J.GuoJ.Guo
4579
asked Feb 2 at 23:10
J.GuoJ.Guo
4579
asked Feb 2 at 23:10
J.GuoJ.Guo
4579
4579
$begingroup$
See en.wikipedia.org/wiki/Harmonic_function
$endgroup$
– Kavi Rama Murthy
Feb 2 at 23:13
add a comment |
$begingroup$
See en.wikipedia.org/wiki/Harmonic_function
$endgroup$
– Kavi Rama Murthy
Feb 2 at 23:13
$begingroup$
See en.wikipedia.org/wiki/Harmonic_function
$endgroup$
– Kavi Rama Murthy
Feb 2 at 23:13
$begingroup$
See en.wikipedia.org/wiki/Harmonic_function
$endgroup$
– Kavi Rama Murthy
Feb 2 at 23:13
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%2f3097920%2fwhy-the-definition-of-harmonic-function-needs-twice-continuously-differentia%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%2f3097920%2fwhy-the-definition-of-harmonic-function-needs-twice-continuously-differentia%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
$begingroup$
See en.wikipedia.org/wiki/Harmonic_function
$endgroup$
– Kavi Rama Murthy
Feb 2 at 23:13