Prove that $u=v$
$begingroup$
I got the following integral identity
$$int_{Omega}left[H(nabla u)(nabla H)(nabla u)-H(nabla v)(nabla H)(nabla v)right]cdotnablaleft(u-vright);dx=0$$
and i want to prove that $u=v$.
Note that $H$ is a Finsler norm, who is homogeneous of degree 1 and convex. How can I use that? Also, since every two norms are equivalent on $mathbb{R}^N$ there exists a,b>0 so that $a|x|leq H(x)leq b|x|$.
real-analysis calculus functional-analysis pde calculus-of-variations
asked Dec 2 '18 at 17:51
AndrewAndrew
346
$endgroup$
add a comment |
$begingroup$
I got the following integral identity
$$int_{Omega}left[H(nabla u)(nabla H)(nabla u)-H(nabla v)(nabla H)(nabla v)right]cdotnablaleft(u-vright);dx=0$$
and i want to prove that $u=v$.
Note that $H$ is a Finsler norm, who is homogeneous of degree 1 and convex. How can I use that? Also, since every two norms are equivalent on $mathbb{R}^N$ there exists a,b>0 so that $a|x|leq H(x)leq b|x|$.
real-analysis calculus functional-analysis pde calculus-of-variations
asked Dec 2 '18 at 17:51
AndrewAndrew
346
$endgroup$
$begingroup$
You need some kind of boundary condition, otherwise you can't rule out examples of the form $u-v=mathrm{const.}$
$endgroup$
– MaoWao
Dec 5 '18 at 13:15
add a comment |
$begingroup$
I got the following integral identity
$$int_{Omega}left[H(nabla u)(nabla H)(nabla u)-H(nabla v)(nabla H)(nabla v)right]cdotnablaleft(u-vright);dx=0$$
and i want to prove that $u=v$.
Note that $H$ is a Finsler norm, who is homogeneous of degree 1 and convex. How can I use that? Also, since every two norms are equivalent on $mathbb{R}^N$ there exists a,b>0 so that $a|x|leq H(x)leq b|x|$.
real-analysis calculus functional-analysis pde calculus-of-variations
asked Dec 2 '18 at 17:51
AndrewAndrew
346
$endgroup$
I got the following integral identity
$$int_{Omega}left[H(nabla u)(nabla H)(nabla u)-H(nabla v)(nabla H)(nabla v)right]cdotnablaleft(u-vright);dx=0$$
and i want to prove that $u=v$.
Note that $H$ is a Finsler norm, who is homogeneous of degree 1 and convex. How can I use that? Also, since every two norms are equivalent on $mathbb{R}^N$ there exists a,b>0 so that $a|x|leq H(x)leq b|x|$.
real-analysis calculus functional-analysis pde calculus-of-variations
real-analysis calculus functional-analysis pde calculus-of-variations
asked Dec 2 '18 at 17:51
AndrewAndrew
346
asked Dec 2 '18 at 17:51
AndrewAndrew
346
asked Dec 2 '18 at 17:51
AndrewAndrew
346
asked Dec 2 '18 at 17:51
AndrewAndrew
346
asked Dec 2 '18 at 17:51
AndrewAndrew
346
346
$begingroup$
You need some kind of boundary condition, otherwise you can't rule out examples of the form $u-v=mathrm{const.}$
$endgroup$
– MaoWao
Dec 5 '18 at 13:15
add a comment |
$begingroup$
You need some kind of boundary condition, otherwise you can't rule out examples of the form $u-v=mathrm{const.}$
$endgroup$
– MaoWao
Dec 5 '18 at 13:15
$begingroup$
You need some kind of boundary condition, otherwise you can't rule out examples of the form $u-v=mathrm{const.}$
$endgroup$
– MaoWao
Dec 5 '18 at 13:15
$begingroup$
You need some kind of boundary condition, otherwise you can't rule out examples of the form $u-v=mathrm{const.}$
$endgroup$
– MaoWao
Dec 5 '18 at 13:15
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint:
What is the derivative of the (nonlinear) functional
$$
u mapsto frac 12 , int_Omega H(nabla u)^2 , mathrm{d}x
$$
?
answered Dec 3 '18 at 7:26
gerwgerw
19.1k11233
$endgroup$
$begingroup$
$int_{Omega}H(nabla u)(nabla H)(nabla u)$;dx
$endgroup$
– Andrew
Dec 3 '18 at 9:00
$begingroup$
Yes. And you can use this to rewrite your equality.
$endgroup$
– gerw
Dec 3 '18 at 11:36
$begingroup$
Can I use the convexity here? how?
$endgroup$
– Andrew
Dec 3 '18 at 18:43
$begingroup$
Yes, convexity is your friend. You essentially need the monotonicity of the derivative of a convex function.
$endgroup$
– gerw
Dec 3 '18 at 20:18
$begingroup$
Also, I put this as a form on inner product something like $$biglangle J'left(uright)-J'(v),u-vbigrangle=0.$$ But I don't know to prove something like strongly convexity of the functional J. Any hints?
$endgroup$
– Andrew
Dec 3 '18 at 20:35
|
show 2 more comments
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint:
What is the derivative of the (nonlinear) functional
$$
u mapsto frac 12 , int_Omega H(nabla u)^2 , mathrm{d}x
$$
?
answered Dec 3 '18 at 7:26
gerwgerw
19.1k11233
$endgroup$
$begingroup$
$int_{Omega}H(nabla u)(nabla H)(nabla u)$;dx
$endgroup$
– Andrew
Dec 3 '18 at 9:00
$begingroup$
Yes. And you can use this to rewrite your equality.
$endgroup$
– gerw
Dec 3 '18 at 11:36
$begingroup$
Can I use the convexity here? how?
$endgroup$
– Andrew
Dec 3 '18 at 18:43
$begingroup$
Yes, convexity is your friend. You essentially need the monotonicity of the derivative of a convex function.
$endgroup$
– gerw
Dec 3 '18 at 20:18
$begingroup$
Also, I put this as a form on inner product something like $$biglangle J'left(uright)-J'(v),u-vbigrangle=0.$$ But I don't know to prove something like strongly convexity of the functional J. Any hints?
$endgroup$
– Andrew
Dec 3 '18 at 20:35
|
show 2 more comments
$begingroup$
Hint:
What is the derivative of the (nonlinear) functional
$$
u mapsto frac 12 , int_Omega H(nabla u)^2 , mathrm{d}x
$$
?
answered Dec 3 '18 at 7:26
gerwgerw
19.1k11233
$endgroup$
$begingroup$
$int_{Omega}H(nabla u)(nabla H)(nabla u)$;dx
$endgroup$
– Andrew
Dec 3 '18 at 9:00
$begingroup$
Yes. And you can use this to rewrite your equality.
$endgroup$
– gerw
Dec 3 '18 at 11:36
$begingroup$
Can I use the convexity here? how?
$endgroup$
– Andrew
Dec 3 '18 at 18:43
$begingroup$
Yes, convexity is your friend. You essentially need the monotonicity of the derivative of a convex function.
$endgroup$
– gerw
Dec 3 '18 at 20:18
$begingroup$
Also, I put this as a form on inner product something like $$biglangle J'left(uright)-J'(v),u-vbigrangle=0.$$ But I don't know to prove something like strongly convexity of the functional J. Any hints?
$endgroup$
– Andrew
Dec 3 '18 at 20:35
|
show 2 more comments
$begingroup$
Hint:
What is the derivative of the (nonlinear) functional
$$
u mapsto frac 12 , int_Omega H(nabla u)^2 , mathrm{d}x
$$
?
answered Dec 3 '18 at 7:26
gerwgerw
19.1k11233
$endgroup$
Hint:
What is the derivative of the (nonlinear) functional
$$
u mapsto frac 12 , int_Omega H(nabla u)^2 , mathrm{d}x
$$
?
answered Dec 3 '18 at 7:26
gerwgerw
19.1k11233
answered Dec 3 '18 at 7:26
gerwgerw
19.1k11233
answered Dec 3 '18 at 7:26
gerwgerw
19.1k11233
answered Dec 3 '18 at 7:26
gerwgerw
19.1k11233
19.1k11233
$begingroup$
$int_{Omega}H(nabla u)(nabla H)(nabla u)$;dx
$endgroup$
– Andrew
Dec 3 '18 at 9:00
$begingroup$
Yes. And you can use this to rewrite your equality.
$endgroup$
– gerw
Dec 3 '18 at 11:36
$begingroup$
Can I use the convexity here? how?
$endgroup$
– Andrew
Dec 3 '18 at 18:43
$begingroup$
Yes, convexity is your friend. You essentially need the monotonicity of the derivative of a convex function.
$endgroup$
– gerw
Dec 3 '18 at 20:18
$begingroup$
Also, I put this as a form on inner product something like $$biglangle J'left(uright)-J'(v),u-vbigrangle=0.$$ But I don't know to prove something like strongly convexity of the functional J. Any hints?
$endgroup$
– Andrew
Dec 3 '18 at 20:35
|
show 2 more comments
$begingroup$
$int_{Omega}H(nabla u)(nabla H)(nabla u)$;dx
$endgroup$
– Andrew
Dec 3 '18 at 9:00
$begingroup$
Yes. And you can use this to rewrite your equality.
$endgroup$
– gerw
Dec 3 '18 at 11:36
$begingroup$
Can I use the convexity here? how?
$endgroup$
– Andrew
Dec 3 '18 at 18:43
$begingroup$
Yes, convexity is your friend. You essentially need the monotonicity of the derivative of a convex function.
$endgroup$
– gerw
Dec 3 '18 at 20:18
$begingroup$
Also, I put this as a form on inner product something like $$biglangle J'left(uright)-J'(v),u-vbigrangle=0.$$ But I don't know to prove something like strongly convexity of the functional J. Any hints?
$endgroup$
– Andrew
Dec 3 '18 at 20:35
$begingroup$
$int_{Omega}H(nabla u)(nabla H)(nabla u)$;dx
$endgroup$
– Andrew
Dec 3 '18 at 9:00
$begingroup$
$int_{Omega}H(nabla u)(nabla H)(nabla u)$;dx
$endgroup$
– Andrew
Dec 3 '18 at 9:00
$begingroup$
Yes. And you can use this to rewrite your equality.
$endgroup$
– gerw
Dec 3 '18 at 11:36
$begingroup$
Yes. And you can use this to rewrite your equality.
$endgroup$
– gerw
Dec 3 '18 at 11:36
$begingroup$
Can I use the convexity here? how?
$endgroup$
– Andrew
Dec 3 '18 at 18:43
$begingroup$
Can I use the convexity here? how?
$endgroup$
– Andrew
Dec 3 '18 at 18:43
$begingroup$
Yes, convexity is your friend. You essentially need the monotonicity of the derivative of a convex function.
$endgroup$
– gerw
Dec 3 '18 at 20:18
$begingroup$
Yes, convexity is your friend. You essentially need the monotonicity of the derivative of a convex function.
$endgroup$
– gerw
Dec 3 '18 at 20:18
$begingroup$
Also, I put this as a form on inner product something like $$biglangle J'left(uright)-J'(v),u-vbigrangle=0.$$ But I don't know to prove something like strongly convexity of the functional J. Any hints?
$endgroup$
– Andrew
Dec 3 '18 at 20:35
$begingroup$
Also, I put this as a form on inner product something like $$biglangle J'left(uright)-J'(v),u-vbigrangle=0.$$ But I don't know to prove something like strongly convexity of the functional J. Any hints?
$endgroup$
– Andrew
Dec 3 '18 at 20:35
|
show 2 more comments
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$begingroup$
You need some kind of boundary condition, otherwise you can't rule out examples of the form $u-v=mathrm{const.}$
$endgroup$
– MaoWao
Dec 5 '18 at 13:15