What is a smooth function on the closure of an open set?
$begingroup$
Assume $U$ is an open subset of $R^{n}$. What does the notation $C^{infty}(bar{U})$ mean? For people who are familiar with sheaf, it should mean the functions on $bar{U}$ which are the restrictions of smooth functions on a neighborhood of $bar{U}$. For people familiar with PDE, it always means smooth functions on $U$ of which any derivatives(including the original function) can continuously extend to $bar{U}$. Are these two views the same? For $U$ with smooth boundary, they refer to the same class of functions. But what about general $U$?
calculus derivatives pde sheaf-theory
asked Dec 20 '18 at 10:03
殷晨曦殷晨曦
212
$endgroup$
add a comment |
$begingroup$
Assume $U$ is an open subset of $R^{n}$. What does the notation $C^{infty}(bar{U})$ mean? For people who are familiar with sheaf, it should mean the functions on $bar{U}$ which are the restrictions of smooth functions on a neighborhood of $bar{U}$. For people familiar with PDE, it always means smooth functions on $U$ of which any derivatives(including the original function) can continuously extend to $bar{U}$. Are these two views the same? For $U$ with smooth boundary, they refer to the same class of functions. But what about general $U$?
calculus derivatives pde sheaf-theory
asked Dec 20 '18 at 10:03
殷晨曦殷晨曦
212
$endgroup$
$begingroup$
That $fin mathcal C^infty (U)$ and for all $uin partial U$ (the boundary of $U$) and all $alpha $ (multi indice), $lim_{xto u}partial _alpha f(x)$ exist.
$endgroup$
– Surb
Dec 20 '18 at 10:06
add a comment |
$begingroup$
Assume $U$ is an open subset of $R^{n}$. What does the notation $C^{infty}(bar{U})$ mean? For people who are familiar with sheaf, it should mean the functions on $bar{U}$ which are the restrictions of smooth functions on a neighborhood of $bar{U}$. For people familiar with PDE, it always means smooth functions on $U$ of which any derivatives(including the original function) can continuously extend to $bar{U}$. Are these two views the same? For $U$ with smooth boundary, they refer to the same class of functions. But what about general $U$?
calculus derivatives pde sheaf-theory
asked Dec 20 '18 at 10:03
殷晨曦殷晨曦
212
$endgroup$
Assume $U$ is an open subset of $R^{n}$. What does the notation $C^{infty}(bar{U})$ mean? For people who are familiar with sheaf, it should mean the functions on $bar{U}$ which are the restrictions of smooth functions on a neighborhood of $bar{U}$. For people familiar with PDE, it always means smooth functions on $U$ of which any derivatives(including the original function) can continuously extend to $bar{U}$. Are these two views the same? For $U$ with smooth boundary, they refer to the same class of functions. But what about general $U$?
calculus derivatives pde sheaf-theory
calculus derivatives pde sheaf-theory
asked Dec 20 '18 at 10:03
殷晨曦殷晨曦
212
asked Dec 20 '18 at 10:03
殷晨曦殷晨曦
212
asked Dec 20 '18 at 10:03
殷晨曦殷晨曦
212
asked Dec 20 '18 at 10:03
殷晨曦殷晨曦
212
asked Dec 20 '18 at 10:03
殷晨曦殷晨曦
212
212
$begingroup$
That $fin mathcal C^infty (U)$ and for all $uin partial U$ (the boundary of $U$) and all $alpha $ (multi indice), $lim_{xto u}partial _alpha f(x)$ exist.
$endgroup$
– Surb
Dec 20 '18 at 10:06
add a comment |
$begingroup$
That $fin mathcal C^infty (U)$ and for all $uin partial U$ (the boundary of $U$) and all $alpha $ (multi indice), $lim_{xto u}partial _alpha f(x)$ exist.
$endgroup$
– Surb
Dec 20 '18 at 10:06
$begingroup$
That $fin mathcal C^infty (U)$ and for all $uin partial U$ (the boundary of $U$) and all $alpha $ (multi indice), $lim_{xto u}partial _alpha f(x)$ exist.
$endgroup$
– Surb
Dec 20 '18 at 10:06
$begingroup$
That $fin mathcal C^infty (U)$ and for all $uin partial U$ (the boundary of $U$) and all $alpha $ (multi indice), $lim_{xto u}partial _alpha f(x)$ exist.
$endgroup$
– Surb
Dec 20 '18 at 10:06
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%2f3047362%2fwhat-is-a-smooth-function-on-the-closure-of-an-open-set%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%2f3047362%2fwhat-is-a-smooth-function-on-the-closure-of-an-open-set%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
$begingroup$
That $fin mathcal C^infty (U)$ and for all $uin partial U$ (the boundary of $U$) and all $alpha $ (multi indice), $lim_{xto u}partial _alpha f(x)$ exist.
$endgroup$
– Surb
Dec 20 '18 at 10:06