How to prove this result somewhat similar to Du Bois-Reymond's Lemma?
up vote
0
down vote
favorite
Let $Omega subseteq mathbb{R}^n$ be an open bounded connected set with smooth boundary. Suppose also that for each index $i, j = 1, dots, n$, $f_{ij}$ is a smooth function.
If for every $vin C^1(overline{Omega})$ there holds
$$
int_{partialOmega} sum_{i,j=1}^n f_{ij}v nu^j = 0
$$
where $nu$ denotes the normal vector then can we conclude that $sum_{i,j=1}^n f_{ij}nu^i = 0$ on $partialOmega$?
I tried "flatening out the boundary" near a point and the applying Du Bois-Reymond's lemma. But what I get doesn't actually fulfill all the conditions to use Du Bois-Reymond's lemma.
If someone can think of another approach that's worth trying, I appreciate the input!
real-analysis functional-analysis lebesgue-integral weak-convergence
asked yesterday
MathUser_NotPrime
968112
add a comment |
up vote
0
down vote
favorite
Let $Omega subseteq mathbb{R}^n$ be an open bounded connected set with smooth boundary. Suppose also that for each index $i, j = 1, dots, n$, $f_{ij}$ is a smooth function.
If for every $vin C^1(overline{Omega})$ there holds
$$
int_{partialOmega} sum_{i,j=1}^n f_{ij}v nu^j = 0
$$
where $nu$ denotes the normal vector then can we conclude that $sum_{i,j=1}^n f_{ij}nu^i = 0$ on $partialOmega$?
I tried "flatening out the boundary" near a point and the applying Du Bois-Reymond's lemma. But what I get doesn't actually fulfill all the conditions to use Du Bois-Reymond's lemma.
If someone can think of another approach that's worth trying, I appreciate the input!
real-analysis functional-analysis lebesgue-integral weak-convergence
asked yesterday
MathUser_NotPrime
968112
Do you use Einstein's summing convention?
– gerw
yesterday
@gerw Yes! Sorry, I should have mentioned that.
– MathUser_NotPrime
yesterday
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $Omega subseteq mathbb{R}^n$ be an open bounded connected set with smooth boundary. Suppose also that for each index $i, j = 1, dots, n$, $f_{ij}$ is a smooth function.
If for every $vin C^1(overline{Omega})$ there holds
$$
int_{partialOmega} sum_{i,j=1}^n f_{ij}v nu^j = 0
$$
where $nu$ denotes the normal vector then can we conclude that $sum_{i,j=1}^n f_{ij}nu^i = 0$ on $partialOmega$?
I tried "flatening out the boundary" near a point and the applying Du Bois-Reymond's lemma. But what I get doesn't actually fulfill all the conditions to use Du Bois-Reymond's lemma.
If someone can think of another approach that's worth trying, I appreciate the input!
real-analysis functional-analysis lebesgue-integral weak-convergence
asked yesterday
MathUser_NotPrime
968112
Let $Omega subseteq mathbb{R}^n$ be an open bounded connected set with smooth boundary. Suppose also that for each index $i, j = 1, dots, n$, $f_{ij}$ is a smooth function.
If for every $vin C^1(overline{Omega})$ there holds
$$
int_{partialOmega} sum_{i,j=1}^n f_{ij}v nu^j = 0
$$
where $nu$ denotes the normal vector then can we conclude that $sum_{i,j=1}^n f_{ij}nu^i = 0$ on $partialOmega$?
I tried "flatening out the boundary" near a point and the applying Du Bois-Reymond's lemma. But what I get doesn't actually fulfill all the conditions to use Du Bois-Reymond's lemma.
If someone can think of another approach that's worth trying, I appreciate the input!
real-analysis functional-analysis lebesgue-integral weak-convergence
real-analysis functional-analysis lebesgue-integral weak-convergence
asked yesterday
MathUser_NotPrime
968112
asked yesterday
MathUser_NotPrime
968112
edited yesterday
asked yesterday
MathUser_NotPrime
968112
asked yesterday
MathUser_NotPrime
968112
asked yesterday
MathUser_NotPrime
968112
968112
Do you use Einstein's summing convention?
– gerw
yesterday
@gerw Yes! Sorry, I should have mentioned that.
– MathUser_NotPrime
yesterday
add a comment |
Do you use Einstein's summing convention?
– gerw
yesterday
@gerw Yes! Sorry, I should have mentioned that.
– MathUser_NotPrime
yesterday
Do you use Einstein's summing convention?
– gerw
yesterday
Do you use Einstein's summing convention?
– gerw
yesterday
@gerw Yes! Sorry, I should have mentioned that.
– MathUser_NotPrime
yesterday
@gerw Yes! Sorry, I should have mentioned that.
– MathUser_NotPrime
yesterday
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f2997848%2fhow-to-prove-this-result-somewhat-similar-to-du-bois-reymonds-lemma%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
Do you use Einstein's summing convention?
– gerw
yesterday
@gerw Yes! Sorry, I should have mentioned that.
– MathUser_NotPrime
yesterday