Let $X={uin C^1[0,1]|u(0)=0}$ and let $I:Xtomathbb{R}$ be defined as $I(u)=int_0^1 (u'^2-u^2)$. Which of the...
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Let $X={uin C^1[0,1]|u(0)=0}$ and let $I:Xtomathbb{R}$ be defined as $I(u)=int_0^1 (u'^2-u^2)$. Which of the following is correct?
$(a)$ I is bounded below
$(b)$ I is not bounded below
$(c)$ I attains its infimum
$(d)$ I does not attain its infimum
Attempt:
Using Euler-Lagrange equations,
$2u+2u'=0$
Therefore $u(x)=c_1sin(x)+c_2 cos(x)$
It is given that $u(0)=0$ so it implies that $c_2=0$.
$therefore u(x)=c_1sin(x)$
What should be the next step? Please give some hints.
calculus calculus-of-variations
asked Nov 15 at 9:28
StammeringMathematician
2,1661322
add a comment |
up vote
0
down vote
favorite
Let $X={uin C^1[0,1]|u(0)=0}$ and let $I:Xtomathbb{R}$ be defined as $I(u)=int_0^1 (u'^2-u^2)$. Which of the following is correct?
$(a)$ I is bounded below
$(b)$ I is not bounded below
$(c)$ I attains its infimum
$(d)$ I does not attain its infimum
Attempt:
Using Euler-Lagrange equations,
$2u+2u'=0$
Therefore $u(x)=c_1sin(x)+c_2 cos(x)$
It is given that $u(0)=0$ so it implies that $c_2=0$.
$therefore u(x)=c_1sin(x)$
What should be the next step? Please give some hints.
calculus calculus-of-variations
asked Nov 15 at 9:28
StammeringMathematician
2,1661322
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $X={uin C^1[0,1]|u(0)=0}$ and let $I:Xtomathbb{R}$ be defined as $I(u)=int_0^1 (u'^2-u^2)$. Which of the following is correct?
$(a)$ I is bounded below
$(b)$ I is not bounded below
$(c)$ I attains its infimum
$(d)$ I does not attain its infimum
Attempt:
Using Euler-Lagrange equations,
$2u+2u'=0$
Therefore $u(x)=c_1sin(x)+c_2 cos(x)$
It is given that $u(0)=0$ so it implies that $c_2=0$.
$therefore u(x)=c_1sin(x)$
What should be the next step? Please give some hints.
calculus calculus-of-variations
asked Nov 15 at 9:28
StammeringMathematician
2,1661322
Let $X={uin C^1[0,1]|u(0)=0}$ and let $I:Xtomathbb{R}$ be defined as $I(u)=int_0^1 (u'^2-u^2)$. Which of the following is correct?
$(a)$ I is bounded below
$(b)$ I is not bounded below
$(c)$ I attains its infimum
$(d)$ I does not attain its infimum
Attempt:
Using Euler-Lagrange equations,
$2u+2u'=0$
Therefore $u(x)=c_1sin(x)+c_2 cos(x)$
It is given that $u(0)=0$ so it implies that $c_2=0$.
$therefore u(x)=c_1sin(x)$
What should be the next step? Please give some hints.
calculus calculus-of-variations
calculus calculus-of-variations
asked Nov 15 at 9:28
StammeringMathematician
2,1661322
asked Nov 15 at 9:28
StammeringMathematician
2,1661322
asked Nov 15 at 9:28
StammeringMathematician
2,1661322
asked Nov 15 at 9:28
StammeringMathematician
2,1661322
asked Nov 15 at 9:28
StammeringMathematician
2,1661322
2,1661322
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
$u(x) =int_0^{x} u'(t) , dt$ so $u(x)^{2} leq xint_0^{x} u'(t) ^{2} , dt leq int_0^{1} u'(t) ^{2} , dt$ which shows (after integration) that $I geq 0$. Hence a) is true, b) is false. The infimum is attained when $u equiv 0$ so c) is true and d) is false.
answered Nov 15 at 9:33
Kavi Rama Murthy
39.9k31750
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
$u(x) =int_0^{x} u'(t) , dt$ so $u(x)^{2} leq xint_0^{x} u'(t) ^{2} , dt leq int_0^{1} u'(t) ^{2} , dt$ which shows (after integration) that $I geq 0$. Hence a) is true, b) is false. The infimum is attained when $u equiv 0$ so c) is true and d) is false.
answered Nov 15 at 9:33
Kavi Rama Murthy
39.9k31750
add a comment |
up vote
2
down vote
accepted
$u(x) =int_0^{x} u'(t) , dt$ so $u(x)^{2} leq xint_0^{x} u'(t) ^{2} , dt leq int_0^{1} u'(t) ^{2} , dt$ which shows (after integration) that $I geq 0$. Hence a) is true, b) is false. The infimum is attained when $u equiv 0$ so c) is true and d) is false.
answered Nov 15 at 9:33
Kavi Rama Murthy
39.9k31750
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
$u(x) =int_0^{x} u'(t) , dt$ so $u(x)^{2} leq xint_0^{x} u'(t) ^{2} , dt leq int_0^{1} u'(t) ^{2} , dt$ which shows (after integration) that $I geq 0$. Hence a) is true, b) is false. The infimum is attained when $u equiv 0$ so c) is true and d) is false.
answered Nov 15 at 9:33
Kavi Rama Murthy
39.9k31750
$u(x) =int_0^{x} u'(t) , dt$ so $u(x)^{2} leq xint_0^{x} u'(t) ^{2} , dt leq int_0^{1} u'(t) ^{2} , dt$ which shows (after integration) that $I geq 0$. Hence a) is true, b) is false. The infimum is attained when $u equiv 0$ so c) is true and d) is false.
answered Nov 15 at 9:33
Kavi Rama Murthy
39.9k31750
answered Nov 15 at 9:33
Kavi Rama Murthy
39.9k31750
answered Nov 15 at 9:33
Kavi Rama Murthy
39.9k31750
answered Nov 15 at 9:33
Kavi Rama Murthy
39.9k31750
39.9k31750
add a comment |
add a comment |
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