Integral inequality with partial derivative
$begingroup$
Fix $eta in Bbb R^n$, then for any $phi in C_c^infty(Bbb R^n)$, i.e. compactly supported smooth function,
$$ int_{Bbb R^n} frac{partial}{partial x_j} (phi(x) (x eta)) , dx le int_{Bbb R^n} |eta| Big| frac{partial}{partial x_j} (phi(x)x) Big| , dx $$
Here $x eta $ denotes dot product.
Any hints/ advice?
EDIT: I think this is correct.
Consider the bounded operator, $L_eta:Bbb R^n rightarrow Bbb R^n$, $x mapsto langle x, eta rangle $. Its norm is given by $||eta||_2$ by Hilbert space duality.By continuity, we have
$$| frac{partial}{partial x_j} L_eta (phi(x)x) | = | L_eta(frac{partial}{partial x_j} phi(x)x ) | le ||L_eta|| , || frac{partial}{partial x_j} phi(x)x ||_2 = ||eta||_2 || frac{partial}{partial x_j} phi(x)x ||_2 $$
real-analysis inequality integral-inequality
asked Dec 23 '18 at 8:24
CL.CL.
2,2882925
$endgroup$
add a comment |
$begingroup$
Fix $eta in Bbb R^n$, then for any $phi in C_c^infty(Bbb R^n)$, i.e. compactly supported smooth function,
$$ int_{Bbb R^n} frac{partial}{partial x_j} (phi(x) (x eta)) , dx le int_{Bbb R^n} |eta| Big| frac{partial}{partial x_j} (phi(x)x) Big| , dx $$
Here $x eta $ denotes dot product.
Any hints/ advice?
EDIT: I think this is correct.
Consider the bounded operator, $L_eta:Bbb R^n rightarrow Bbb R^n$, $x mapsto langle x, eta rangle $. Its norm is given by $||eta||_2$ by Hilbert space duality.By continuity, we have
$$| frac{partial}{partial x_j} L_eta (phi(x)x) | = | L_eta(frac{partial}{partial x_j} phi(x)x ) | le ||L_eta|| , || frac{partial}{partial x_j} phi(x)x ||_2 = ||eta||_2 || frac{partial}{partial x_j} phi(x)x ||_2 $$
real-analysis inequality integral-inequality
asked Dec 23 '18 at 8:24
CL.CL.
2,2882925
$endgroup$
add a comment |
$begingroup$
Fix $eta in Bbb R^n$, then for any $phi in C_c^infty(Bbb R^n)$, i.e. compactly supported smooth function,
$$ int_{Bbb R^n} frac{partial}{partial x_j} (phi(x) (x eta)) , dx le int_{Bbb R^n} |eta| Big| frac{partial}{partial x_j} (phi(x)x) Big| , dx $$
Here $x eta $ denotes dot product.
Any hints/ advice?
EDIT: I think this is correct.
Consider the bounded operator, $L_eta:Bbb R^n rightarrow Bbb R^n$, $x mapsto langle x, eta rangle $. Its norm is given by $||eta||_2$ by Hilbert space duality.By continuity, we have
$$| frac{partial}{partial x_j} L_eta (phi(x)x) | = | L_eta(frac{partial}{partial x_j} phi(x)x ) | le ||L_eta|| , || frac{partial}{partial x_j} phi(x)x ||_2 = ||eta||_2 || frac{partial}{partial x_j} phi(x)x ||_2 $$
real-analysis inequality integral-inequality
asked Dec 23 '18 at 8:24
CL.CL.
2,2882925
$endgroup$
Fix $eta in Bbb R^n$, then for any $phi in C_c^infty(Bbb R^n)$, i.e. compactly supported smooth function,
$$ int_{Bbb R^n} frac{partial}{partial x_j} (phi(x) (x eta)) , dx le int_{Bbb R^n} |eta| Big| frac{partial}{partial x_j} (phi(x)x) Big| , dx $$
Here $x eta $ denotes dot product.
Any hints/ advice?
EDIT: I think this is correct.
Consider the bounded operator, $L_eta:Bbb R^n rightarrow Bbb R^n$, $x mapsto langle x, eta rangle $. Its norm is given by $||eta||_2$ by Hilbert space duality.By continuity, we have
$$| frac{partial}{partial x_j} L_eta (phi(x)x) | = | L_eta(frac{partial}{partial x_j} phi(x)x ) | le ||L_eta|| , || frac{partial}{partial x_j} phi(x)x ||_2 = ||eta||_2 || frac{partial}{partial x_j} phi(x)x ||_2 $$
real-analysis inequality integral-inequality
real-analysis inequality integral-inequality
asked Dec 23 '18 at 8:24
CL.CL.
2,2882925
asked Dec 23 '18 at 8:24
CL.CL.
2,2882925
edited Dec 23 '18 at 8:34
CL.
asked Dec 23 '18 at 8:24
CL.CL.
2,2882925
asked Dec 23 '18 at 8:24
CL.CL.
2,2882925
asked Dec 23 '18 at 8:24
CL.CL.
2,2882925
2,2882925
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%2f3050164%2fintegral-inequality-with-partial-derivative%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%2f3050164%2fintegral-inequality-with-partial-derivative%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
