Fdtxjr

Intuition behind counterexample of Euler's sum of powers conjecture

Bash script should only kill those instances of another script's that it has launched

In the late 1940’s to early 1950’s what technology was available that could melt a LOT of ice?

How to draw cubes in a 3 dimensional plane

How to secure an aircraft at a transient parking space?

Does the nature of the Apocalypse in The Umbrella Academy change from the first to the last episode?

Should I take out a loan for a friend to invest on my behalf?

Virginia employer terminated employee and wants signing bonus returned

Does a warlock using the Darkness/Devil's Sight combo still have advantage on ranged attacks against a target outside the Darkness?

Definition of Statistic

Is it possible to avoid unpacking when merging Association?

Does "Until when" sound natural for native speakers?

How can I ensure my trip to the UK will not have to be cancelled because of Brexit?

Signed and unsigned numbers

Declaring and defining template, and specialising them

Is it "Vierergruppe" or "Viergruppe", or is there a distinction?

Good for you! in Russian

List elements digit difference sort

Are babies of evil humanoid species inherently evil?

weren't playing vs didn't play

Do f-stop and exposure time perfectly cancel?

How many characters using PHB rules does it take to be able to have access to any PHB spell at the start of an adventuring day?

How is the wildcard * interpreted as a command?

Contract Factories

Monotonicty of the trace on Sobolev spaces.

When is it true that the Sobolev trace of a positive a.e. function is positive a.e?Sobolev inequality in $W_0^1,p$How to prove that $(u-v)^+in W_0^1,2(Omega)$, if $uin W_0^1,2(Omega)$, $vgeq 0$.About the image of the trace operator for Sobolev spaces.Are all functions in the Sobolev space $W_0^1,2(Omega)$ continuous and bounded?This Sobolev function is continuous?Negative index Sobolev spaces, propertiesCan I conclude that this Sobolev function is Lipschitz?Coincidence of a weak limit and a uniform limit in Sobolev spacesmaximum of trace operator and 0Compact Imbedding on weighted Sobolev Spaces

2

3

$begingroup$

Let $Omega subseteq mathbbR^n$ be a bounded domain with smooth boundary and suppose $uin W_0^1,p(Omega)$. If $vin W^1,p(Omega)$ satisfies $lvert v rvert leq lvert u rvert$ in all of $Omega$, can we conclude that $vin W_0^1,p(Omega)$.

Clearly, the statement is true if we assume that $u$ and $v$ are continuous up to the boundary of $Omega$. More generally, the statement "feels" true and almost intuitive, but I have no idea how to solve or approach the problem.

Thanks in advance for any idea or references about this problem!

real-analysis functional-analysis sobolev-spaces trace

share|cite|improve this question

edited 2 days ago

Quoka

asked Mar 6 at 3:14

QuokaQuoka

1,331313

$endgroup$

add a comment | 

2

3

$begingroup$

Let $Omega subseteq mathbbR^n$ be a bounded domain with smooth boundary and suppose $uin W_0^1,p(Omega)$. If $vin W^1,p(Omega)$ satisfies $lvert v rvert leq lvert u rvert$ in all of $Omega$, can we conclude that $vin W_0^1,p(Omega)$.

Clearly, the statement is true if we assume that $u$ and $v$ are continuous up to the boundary of $Omega$. More generally, the statement "feels" true and almost intuitive, but I have no idea how to solve or approach the problem.

Thanks in advance for any idea or references about this problem!

real-analysis functional-analysis sobolev-spaces trace

share|cite|improve this question

edited 2 days ago

Quoka

asked Mar 6 at 3:14

QuokaQuoka

1,331313

$endgroup$

add a comment | 

$begingroup$

Let $Omega subseteq mathbbR^n$ be a bounded domain with smooth boundary and suppose $uin W_0^1,p(Omega)$. If $vin W^1,p(Omega)$ satisfies $lvert v rvert leq lvert u rvert$ in all of $Omega$, can we conclude that $vin W_0^1,p(Omega)$.

Clearly, the statement is true if we assume that $u$ and $v$ are continuous up to the boundary of $Omega$. More generally, the statement "feels" true and almost intuitive, but I have no idea how to solve or approach the problem.

Thanks in advance for any idea or references about this problem!

real-analysis functional-analysis sobolev-spaces trace

share|cite|improve this question

edited 2 days ago

Quoka

asked Mar 6 at 3:14

QuokaQuoka

1,331313

$endgroup$

share|cite|improve this question

edited 2 days ago

Quoka

asked Mar 6 at 3:14

QuokaQuoka

1,331313

share|cite|improve this question

edited 2 days ago

Quoka

asked Mar 6 at 3:14

QuokaQuoka

1,331313

asked Mar 6 at 3:14

QuokaQuoka

1,331313

asked Mar 6 at 3:14

QuokaQuoka

1,331313

1 Answer
1

active

oldest

votes

5

$begingroup$

Firstly, we will use that there for $ugeq0Rightarrow tr(u)geq0$. See here
for ideas on a proof.

Secondly, we will use that for $uin W^1,p$ also $u^+=max(0,u)in W^1,p$ and similarly $u^-=min(0,u)in W^1,p$.

Thirdly, we will use the trace Operator which is a continuous linear Operator from $W^1,pto L^p$ which allows the following identification $W^1,p_0=uin W^1,p$.

Now, we start by splitting the functions into
$$
u=u^++u^-\
v=v^++v^-\
|v|leq|u|Leftrightarrow v^+leq u^+wedge u^-leq v^-
$$

Now since
$$0leq tr(u^+-v^+)=tr(u^+)-tr(v^+)=-tr(v^+)leq0$$
and
$$0leq tr(v^--u^-)=tr(v^-)-tr(u^-)=tr(v^-)leq0$$
we can deduct that
$$tr(u)=tr(u^++u^-)=tr(u^+)+tr(u^-)=0$$

share|cite|improve this answer

answered Mar 6 at 7:27

maxmilgrammaxmilgram

7607

$endgroup$

add a comment | 

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
);



);

draft saved

draft discarded

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

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3137031%2fmonotonicty-of-the-trace-on-sobolev-spaces%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

5

$begingroup$

Firstly, we will use that there for $ugeq0Rightarrow tr(u)geq0$. See here
for ideas on a proof.

Secondly, we will use that for $uin W^1,p$ also $u^+=max(0,u)in W^1,p$ and similarly $u^-=min(0,u)in W^1,p$.

Thirdly, we will use the trace Operator which is a continuous linear Operator from $W^1,pto L^p$ which allows the following identification $W^1,p_0=uin W^1,p$.

Now, we start by splitting the functions into
$$
u=u^++u^-\
v=v^++v^-\
|v|leq|u|Leftrightarrow v^+leq u^+wedge u^-leq v^-
$$

Now since
$$0leq tr(u^+-v^+)=tr(u^+)-tr(v^+)=-tr(v^+)leq0$$
and
$$0leq tr(v^--u^-)=tr(v^-)-tr(u^-)=tr(v^-)leq0$$
we can deduct that
$$tr(u)=tr(u^++u^-)=tr(u^+)+tr(u^-)=0$$

share|cite|improve this answer

answered Mar 6 at 7:27

maxmilgrammaxmilgram

7607

$endgroup$

add a comment | 

5

$begingroup$

Firstly, we will use that there for $ugeq0Rightarrow tr(u)geq0$. See here
for ideas on a proof.

Secondly, we will use that for $uin W^1,p$ also $u^+=max(0,u)in W^1,p$ and similarly $u^-=min(0,u)in W^1,p$.

Thirdly, we will use the trace Operator which is a continuous linear Operator from $W^1,pto L^p$ which allows the following identification $W^1,p_0=uin W^1,p$.

Now, we start by splitting the functions into
$$
u=u^++u^-\
v=v^++v^-\
|v|leq|u|Leftrightarrow v^+leq u^+wedge u^-leq v^-
$$

Now since
$$0leq tr(u^+-v^+)=tr(u^+)-tr(v^+)=-tr(v^+)leq0$$
and
$$0leq tr(v^--u^-)=tr(v^-)-tr(u^-)=tr(v^-)leq0$$
we can deduct that
$$tr(u)=tr(u^++u^-)=tr(u^+)+tr(u^-)=0$$

share|cite|improve this answer

answered Mar 6 at 7:27

maxmilgrammaxmilgram

7607

$endgroup$

add a comment | 

$begingroup$

Firstly, we will use that there for $ugeq0Rightarrow tr(u)geq0$. See here
for ideas on a proof.

Secondly, we will use that for $uin W^1,p$ also $u^+=max(0,u)in W^1,p$ and similarly $u^-=min(0,u)in W^1,p$.

Thirdly, we will use the trace Operator which is a continuous linear Operator from $W^1,pto L^p$ which allows the following identification $W^1,p_0=uin W^1,p$.

Now, we start by splitting the functions into
$$
u=u^++u^-\
v=v^++v^-\
|v|leq|u|Leftrightarrow v^+leq u^+wedge u^-leq v^-
$$

Now since
$$0leq tr(u^+-v^+)=tr(u^+)-tr(v^+)=-tr(v^+)leq0$$
and
$$0leq tr(v^--u^-)=tr(v^-)-tr(u^-)=tr(v^-)leq0$$
we can deduct that
$$tr(u)=tr(u^++u^-)=tr(u^+)+tr(u^-)=0$$

share|cite|improve this answer

answered Mar 6 at 7:27

maxmilgrammaxmilgram

7607

$endgroup$

share|cite|improve this answer

answered Mar 6 at 7:27

maxmilgrammaxmilgram

7607

answered Mar 6 at 7:27

maxmilgrammaxmilgram

7607

answered Mar 6 at 7:27

maxmilgrammaxmilgram

7607