prove $rhoin Kto ||sqrt{rho}'||_{L^2}^2$ is convex, where $K$ is a convex cone
$begingroup$
$$K = {rhoin H^1(0,1);rho ge 0,sqrt{rho}in H^1(0,1)}$$
I have proved that $K$ is a convex cone. Now I'm asked to prove that $rhoin Kto ||sqrt{rho}'||_{L^2}^2$ is convex.
I tried several ways to compute, but I can only get
$$||sqrt{tu+(1-t)v}'||_{L^2}^2 le 2t||sqrt{u}'||_{L^2}^2 + 2(1-t)||sqrt{v}'||_{L^2}^2 text{ or }le sqrt{t}||sqrt{u}'||_{L^2}^2 + sqrt{1-t}||sqrt{v}'||_{L^2}^2$$
So any ideas how to get convexity?
functional-analysis sobolev-spaces
asked Dec 3 '18 at 21:05
QD666QD666
1346
$endgroup$
add a comment |
$begingroup$
$$K = {rhoin H^1(0,1);rho ge 0,sqrt{rho}in H^1(0,1)}$$
I have proved that $K$ is a convex cone. Now I'm asked to prove that $rhoin Kto ||sqrt{rho}'||_{L^2}^2$ is convex.
I tried several ways to compute, but I can only get
$$||sqrt{tu+(1-t)v}'||_{L^2}^2 le 2t||sqrt{u}'||_{L^2}^2 + 2(1-t)||sqrt{v}'||_{L^2}^2 text{ or }le sqrt{t}||sqrt{u}'||_{L^2}^2 + sqrt{1-t}||sqrt{v}'||_{L^2}^2$$
So any ideas how to get convexity?
functional-analysis sobolev-spaces
asked Dec 3 '18 at 21:05
QD666QD666
1346
$endgroup$
add a comment |
$begingroup$
$$K = {rhoin H^1(0,1);rho ge 0,sqrt{rho}in H^1(0,1)}$$
I have proved that $K$ is a convex cone. Now I'm asked to prove that $rhoin Kto ||sqrt{rho}'||_{L^2}^2$ is convex.
I tried several ways to compute, but I can only get
$$||sqrt{tu+(1-t)v}'||_{L^2}^2 le 2t||sqrt{u}'||_{L^2}^2 + 2(1-t)||sqrt{v}'||_{L^2}^2 text{ or }le sqrt{t}||sqrt{u}'||_{L^2}^2 + sqrt{1-t}||sqrt{v}'||_{L^2}^2$$
So any ideas how to get convexity?
functional-analysis sobolev-spaces
asked Dec 3 '18 at 21:05
QD666QD666
1346
$endgroup$
$$K = {rhoin H^1(0,1);rho ge 0,sqrt{rho}in H^1(0,1)}$$
I have proved that $K$ is a convex cone. Now I'm asked to prove that $rhoin Kto ||sqrt{rho}'||_{L^2}^2$ is convex.
I tried several ways to compute, but I can only get
$$||sqrt{tu+(1-t)v}'||_{L^2}^2 le 2t||sqrt{u}'||_{L^2}^2 + 2(1-t)||sqrt{v}'||_{L^2}^2 text{ or }le sqrt{t}||sqrt{u}'||_{L^2}^2 + sqrt{1-t}||sqrt{v}'||_{L^2}^2$$
So any ideas how to get convexity?
functional-analysis sobolev-spaces
functional-analysis sobolev-spaces
asked Dec 3 '18 at 21:05
QD666QD666
1346
asked Dec 3 '18 at 21:05
QD666QD666
1346
edited Dec 3 '18 at 21:27
QD666
asked Dec 3 '18 at 21:05
QD666QD666
1346
asked Dec 3 '18 at 21:05
QD666QD666
1346
asked Dec 3 '18 at 21:05
QD666QD666
1346
1346
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hints:
- Compute $sqrt{rho}'$.
- Use this to get a nice (integral) expression for $|sqrt{rho}'|_{L^2}^2$.
- Prove that $mathbb R^2 ni (a,b) mapsto a^2/b$ is convex.
answered Dec 4 '18 at 7:09
gerwgerw
19.6k11334
$endgroup$
$begingroup$
yes, my steps are exactly the same as yours, but I'm having trouble to prove the last step
$endgroup$
– QD666
Dec 4 '18 at 19:46
$begingroup$
Which last step? The convexity of $(a,b) mapsto a^2/b$? Just compute its Hessian.
$endgroup$
– gerw
Dec 4 '18 at 20:56
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hints:
- Compute $sqrt{rho}'$.
- Use this to get a nice (integral) expression for $|sqrt{rho}'|_{L^2}^2$.
- Prove that $mathbb R^2 ni (a,b) mapsto a^2/b$ is convex.
answered Dec 4 '18 at 7:09
gerwgerw
19.6k11334
$endgroup$
$begingroup$
yes, my steps are exactly the same as yours, but I'm having trouble to prove the last step
$endgroup$
– QD666
Dec 4 '18 at 19:46
$begingroup$
Which last step? The convexity of $(a,b) mapsto a^2/b$? Just compute its Hessian.
$endgroup$
– gerw
Dec 4 '18 at 20:56
add a comment |
$begingroup$
Hints:
- Compute $sqrt{rho}'$.
- Use this to get a nice (integral) expression for $|sqrt{rho}'|_{L^2}^2$.
- Prove that $mathbb R^2 ni (a,b) mapsto a^2/b$ is convex.
answered Dec 4 '18 at 7:09
gerwgerw
19.6k11334
$endgroup$
$begingroup$
yes, my steps are exactly the same as yours, but I'm having trouble to prove the last step
$endgroup$
– QD666
Dec 4 '18 at 19:46
$begingroup$
Which last step? The convexity of $(a,b) mapsto a^2/b$? Just compute its Hessian.
$endgroup$
– gerw
Dec 4 '18 at 20:56
add a comment |
$begingroup$
Hints:
- Compute $sqrt{rho}'$.
- Use this to get a nice (integral) expression for $|sqrt{rho}'|_{L^2}^2$.
- Prove that $mathbb R^2 ni (a,b) mapsto a^2/b$ is convex.
answered Dec 4 '18 at 7:09
gerwgerw
19.6k11334
$endgroup$
Hints:
- Compute $sqrt{rho}'$.
- Use this to get a nice (integral) expression for $|sqrt{rho}'|_{L^2}^2$.
- Prove that $mathbb R^2 ni (a,b) mapsto a^2/b$ is convex.
answered Dec 4 '18 at 7:09
gerwgerw
19.6k11334
answered Dec 4 '18 at 7:09
gerwgerw
19.6k11334
answered Dec 4 '18 at 7:09
gerwgerw
19.6k11334
answered Dec 4 '18 at 7:09
gerwgerw
19.6k11334
19.6k11334
$begingroup$
yes, my steps are exactly the same as yours, but I'm having trouble to prove the last step
$endgroup$
– QD666
Dec 4 '18 at 19:46
$begingroup$
Which last step? The convexity of $(a,b) mapsto a^2/b$? Just compute its Hessian.
$endgroup$
– gerw
Dec 4 '18 at 20:56
add a comment |
$begingroup$
yes, my steps are exactly the same as yours, but I'm having trouble to prove the last step
$endgroup$
– QD666
Dec 4 '18 at 19:46
$begingroup$
Which last step? The convexity of $(a,b) mapsto a^2/b$? Just compute its Hessian.
$endgroup$
– gerw
Dec 4 '18 at 20:56
$begingroup$
yes, my steps are exactly the same as yours, but I'm having trouble to prove the last step
$endgroup$
– QD666
Dec 4 '18 at 19:46
$begingroup$
yes, my steps are exactly the same as yours, but I'm having trouble to prove the last step
$endgroup$
– QD666
Dec 4 '18 at 19:46
$begingroup$
Which last step? The convexity of $(a,b) mapsto a^2/b$? Just compute its Hessian.
$endgroup$
– gerw
Dec 4 '18 at 20:56
$begingroup$
Which last step? The convexity of $(a,b) mapsto a^2/b$? Just compute its Hessian.
$endgroup$
– gerw
Dec 4 '18 at 20:56
add a comment |
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