Bounded sequence in Sobolev Space
Hi can anyone complete my solution (or give a better solution /hints to a better solution) to the following problem :
Define : The Sobolev Space $W^{1,4}(mathbb{T}^{2})$ as
$$W^{1,4}(mathbb{T}^{2})={u~:~~ uin L^4_{loc}(mathbb{R}^2), ~~ Delta uin L^4_{loc}(mathbb{R}^2),~~ u(x_{1}+1,x_{2})=u(x_{1},x_{2}),~~u(x_{1},x_{2}+1)=u(x_{1}+1,x_{2}),~~ int_{(0,1)^{2}}u~=0 } $$
Define : The functional $J:W^{1,4}(mathbb{T}^2)to mathbb{R} $ (for some $fin L^2(mathbb{T}^2) $)
$$J(u)=frac{1}{2}int_{(0,1)^{2}}|Du|^2 + int_{(0,1)^{2}}u_{x_{1}}^4+u_{x_{2}}^4 - int_{(0,1)^{2}}fu$$
Show that if $J(u_{n})to inf_{u}J(u)$ then ${ u_{n}}$ is bounded in $W^{1,4}(mathbb{T}^2)$.
$bf{Hint}$ Use Poincare-Writinger inequality to justify why $exists C>0$ s.t for any $uin W^{1,4}(mathbb{T}^2)$
$$||u||_{W^{1,4}(mathbb{T}^2)}leq C|| D u||_{W^{1,4}(mathbb{T}^2)} $$
Note : $Du=(u_{x_{1}},u_{x_{2}})$
$bf{My Attempt}$ ( note : we only need to bound the tail of the sequence)
Let $epsilon>0$ using the definition of limit $exists N$ s.t $forall n>N$
$$| J(u_{n})-inf J(u) | leq epsilon $$
Now using definition of infimum it follows
$$J(u_{n})leq epsilon +inf J(u) $$
$$J(u_{n}) leq eta ~~~~ forall n>N $$
Now were done if we can show $||u_{n}||_{W^{1,4}(mathbb{T}^2)} leq J(u_{n})$
Im struggling to extract $||u_{n}||_{W^{1,4}}(mathbb{T}^2)$ from the above because of the $-int_{(0,1)^2} fu_{n} $ term. I guess its a combination of Holder, $f$ zero average, and the 'hint'.
sequences-and-series functional-analysis sobolev-spaces lp-spaces
asked Dec 26 '18 at 17:48
rogerroger
293
New contributor
rogerroger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |
Hi can anyone complete my solution (or give a better solution /hints to a better solution) to the following problem :
Define : The Sobolev Space $W^{1,4}(mathbb{T}^{2})$ as
$$W^{1,4}(mathbb{T}^{2})={u~:~~ uin L^4_{loc}(mathbb{R}^2), ~~ Delta uin L^4_{loc}(mathbb{R}^2),~~ u(x_{1}+1,x_{2})=u(x_{1},x_{2}),~~u(x_{1},x_{2}+1)=u(x_{1}+1,x_{2}),~~ int_{(0,1)^{2}}u~=0 } $$
Define : The functional $J:W^{1,4}(mathbb{T}^2)to mathbb{R} $ (for some $fin L^2(mathbb{T}^2) $)
$$J(u)=frac{1}{2}int_{(0,1)^{2}}|Du|^2 + int_{(0,1)^{2}}u_{x_{1}}^4+u_{x_{2}}^4 - int_{(0,1)^{2}}fu$$
Show that if $J(u_{n})to inf_{u}J(u)$ then ${ u_{n}}$ is bounded in $W^{1,4}(mathbb{T}^2)$.
$bf{Hint}$ Use Poincare-Writinger inequality to justify why $exists C>0$ s.t for any $uin W^{1,4}(mathbb{T}^2)$
$$||u||_{W^{1,4}(mathbb{T}^2)}leq C|| D u||_{W^{1,4}(mathbb{T}^2)} $$
Note : $Du=(u_{x_{1}},u_{x_{2}})$
$bf{My Attempt}$ ( note : we only need to bound the tail of the sequence)
Let $epsilon>0$ using the definition of limit $exists N$ s.t $forall n>N$
$$| J(u_{n})-inf J(u) | leq epsilon $$
Now using definition of infimum it follows
$$J(u_{n})leq epsilon +inf J(u) $$
$$J(u_{n}) leq eta ~~~~ forall n>N $$
Now were done if we can show $||u_{n}||_{W^{1,4}(mathbb{T}^2)} leq J(u_{n})$
Im struggling to extract $||u_{n}||_{W^{1,4}}(mathbb{T}^2)$ from the above because of the $-int_{(0,1)^2} fu_{n} $ term. I guess its a combination of Holder, $f$ zero average, and the 'hint'.
sequences-and-series functional-analysis sobolev-spaces lp-spaces
asked Dec 26 '18 at 17:48
rogerroger
293
New contributor
rogerroger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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What is $Delta$?
– timur
Dec 26 '18 at 19:43
@timur Sorry I meant to write $||Du||_{W^{1,4}(mathbb{T}^2)}$, I will edit now
– rogerroger
Dec 26 '18 at 19:59
add a comment |
Hi can anyone complete my solution (or give a better solution /hints to a better solution) to the following problem :
Define : The Sobolev Space $W^{1,4}(mathbb{T}^{2})$ as
$$W^{1,4}(mathbb{T}^{2})={u~:~~ uin L^4_{loc}(mathbb{R}^2), ~~ Delta uin L^4_{loc}(mathbb{R}^2),~~ u(x_{1}+1,x_{2})=u(x_{1},x_{2}),~~u(x_{1},x_{2}+1)=u(x_{1}+1,x_{2}),~~ int_{(0,1)^{2}}u~=0 } $$
Define : The functional $J:W^{1,4}(mathbb{T}^2)to mathbb{R} $ (for some $fin L^2(mathbb{T}^2) $)
$$J(u)=frac{1}{2}int_{(0,1)^{2}}|Du|^2 + int_{(0,1)^{2}}u_{x_{1}}^4+u_{x_{2}}^4 - int_{(0,1)^{2}}fu$$
Show that if $J(u_{n})to inf_{u}J(u)$ then ${ u_{n}}$ is bounded in $W^{1,4}(mathbb{T}^2)$.
$bf{Hint}$ Use Poincare-Writinger inequality to justify why $exists C>0$ s.t for any $uin W^{1,4}(mathbb{T}^2)$
$$||u||_{W^{1,4}(mathbb{T}^2)}leq C|| D u||_{W^{1,4}(mathbb{T}^2)} $$
Note : $Du=(u_{x_{1}},u_{x_{2}})$
$bf{My Attempt}$ ( note : we only need to bound the tail of the sequence)
Let $epsilon>0$ using the definition of limit $exists N$ s.t $forall n>N$
$$| J(u_{n})-inf J(u) | leq epsilon $$
Now using definition of infimum it follows
$$J(u_{n})leq epsilon +inf J(u) $$
$$J(u_{n}) leq eta ~~~~ forall n>N $$
Now were done if we can show $||u_{n}||_{W^{1,4}(mathbb{T}^2)} leq J(u_{n})$
Im struggling to extract $||u_{n}||_{W^{1,4}}(mathbb{T}^2)$ from the above because of the $-int_{(0,1)^2} fu_{n} $ term. I guess its a combination of Holder, $f$ zero average, and the 'hint'.
sequences-and-series functional-analysis sobolev-spaces lp-spaces
asked Dec 26 '18 at 17:48
rogerroger
293
New contributor
rogerroger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Hi can anyone complete my solution (or give a better solution /hints to a better solution) to the following problem :
Define : The Sobolev Space $W^{1,4}(mathbb{T}^{2})$ as
$$W^{1,4}(mathbb{T}^{2})={u~:~~ uin L^4_{loc}(mathbb{R}^2), ~~ Delta uin L^4_{loc}(mathbb{R}^2),~~ u(x_{1}+1,x_{2})=u(x_{1},x_{2}),~~u(x_{1},x_{2}+1)=u(x_{1}+1,x_{2}),~~ int_{(0,1)^{2}}u~=0 } $$
Define : The functional $J:W^{1,4}(mathbb{T}^2)to mathbb{R} $ (for some $fin L^2(mathbb{T}^2) $)
$$J(u)=frac{1}{2}int_{(0,1)^{2}}|Du|^2 + int_{(0,1)^{2}}u_{x_{1}}^4+u_{x_{2}}^4 - int_{(0,1)^{2}}fu$$
Show that if $J(u_{n})to inf_{u}J(u)$ then ${ u_{n}}$ is bounded in $W^{1,4}(mathbb{T}^2)$.
$bf{Hint}$ Use Poincare-Writinger inequality to justify why $exists C>0$ s.t for any $uin W^{1,4}(mathbb{T}^2)$
$$||u||_{W^{1,4}(mathbb{T}^2)}leq C|| D u||_{W^{1,4}(mathbb{T}^2)} $$
Note : $Du=(u_{x_{1}},u_{x_{2}})$
$bf{My Attempt}$ ( note : we only need to bound the tail of the sequence)
Let $epsilon>0$ using the definition of limit $exists N$ s.t $forall n>N$
$$| J(u_{n})-inf J(u) | leq epsilon $$
Now using definition of infimum it follows
$$J(u_{n})leq epsilon +inf J(u) $$
$$J(u_{n}) leq eta ~~~~ forall n>N $$
Now were done if we can show $||u_{n}||_{W^{1,4}(mathbb{T}^2)} leq J(u_{n})$
Im struggling to extract $||u_{n}||_{W^{1,4}}(mathbb{T}^2)$ from the above because of the $-int_{(0,1)^2} fu_{n} $ term. I guess its a combination of Holder, $f$ zero average, and the 'hint'.
sequences-and-series functional-analysis sobolev-spaces lp-spaces
sequences-and-series functional-analysis sobolev-spaces lp-spaces
asked Dec 26 '18 at 17:48
rogerroger
293
New contributor
rogerroger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 26 '18 at 17:48
rogerroger
293
New contributor
rogerroger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Dec 28 '18 at 23:03
asked Dec 26 '18 at 17:48
rogerroger
293
New contributor
rogerroger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 26 '18 at 17:48
rogerroger
293
asked Dec 26 '18 at 17:48
rogerroger
293
293
New contributor
rogerroger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
rogerroger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
rogerroger is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
What is $Delta$?
– timur
Dec 26 '18 at 19:43
@timur Sorry I meant to write $||Du||_{W^{1,4}(mathbb{T}^2)}$, I will edit now
– rogerroger
Dec 26 '18 at 19:59
add a comment |
What is $Delta$?
– timur
Dec 26 '18 at 19:43
@timur Sorry I meant to write $||Du||_{W^{1,4}(mathbb{T}^2)}$, I will edit now
– rogerroger
Dec 26 '18 at 19:59
What is $Delta$?
– timur
Dec 26 '18 at 19:43
What is $Delta$?
– timur
Dec 26 '18 at 19:43
@timur Sorry I meant to write $||Du||_{W^{1,4}(mathbb{T}^2)}$, I will edit now
– rogerroger
Dec 26 '18 at 19:59
@timur Sorry I meant to write $||Du||_{W^{1,4}(mathbb{T}^2)}$, I will edit now
– rogerroger
Dec 26 '18 at 19:59
add a comment |
1 Answer
1
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oldest
votes
We have
$$
|int fu| leq |f|_{L^2}|u|_{L^2}
leq
varepsilon|u|_{L^2}^2+frac4varepsilon|f|_{L^2}^2
leq
Cvarepsilon|Du|_{L^2}^2+frac4varepsilon|f|_{L^2}^2 ,
$$
where we have used the fact that $int u=0$ in combination with the Poincare-Wirtinger inequality. Take $varepsilon>0$ small enough, and get
$$
|Du|_{L^4}^4leqalpha J(u) + beta
$$
for some constants $alpha,beta$.
answered Dec 26 '18 at 20:55
timur
11.8k1943
Is it obvious that we can apply Poincare-Wirtinger in this way for all $u in W^{1,4}(mathbb{T}^2)$
– rogerroger
Dec 26 '18 at 21:05
Thanks by the way, I should of thought to apply the epsilon young inequality it is used everywhere :)
– rogerroger
Dec 26 '18 at 21:06
1
@rogerroger: Yes, because the domain is bounded, there is the continuous embedding $W^{1,4}subset W^{1,2}$.
– timur
Dec 26 '18 at 21:10
and I assume you have used $u$ having zero average by not regarding it in your $||cdot ||_{W^{1,4}}$ norm?
– rogerroger
Dec 27 '18 at 17:43
I don't know exactly which step you are referring to but you will need that at some point.
– timur
Dec 27 '18 at 19:50
|
show 2 more comments
Your Answer
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1 Answer
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1 Answer
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active
oldest
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oldest
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active
oldest
votes
We have
$$
|int fu| leq |f|_{L^2}|u|_{L^2}
leq
varepsilon|u|_{L^2}^2+frac4varepsilon|f|_{L^2}^2
leq
Cvarepsilon|Du|_{L^2}^2+frac4varepsilon|f|_{L^2}^2 ,
$$
where we have used the fact that $int u=0$ in combination with the Poincare-Wirtinger inequality. Take $varepsilon>0$ small enough, and get
$$
|Du|_{L^4}^4leqalpha J(u) + beta
$$
for some constants $alpha,beta$.
answered Dec 26 '18 at 20:55
timur
11.8k1943
Is it obvious that we can apply Poincare-Wirtinger in this way for all $u in W^{1,4}(mathbb{T}^2)$
– rogerroger
Dec 26 '18 at 21:05
Thanks by the way, I should of thought to apply the epsilon young inequality it is used everywhere :)
– rogerroger
Dec 26 '18 at 21:06
1
@rogerroger: Yes, because the domain is bounded, there is the continuous embedding $W^{1,4}subset W^{1,2}$.
– timur
Dec 26 '18 at 21:10
and I assume you have used $u$ having zero average by not regarding it in your $||cdot ||_{W^{1,4}}$ norm?
– rogerroger
Dec 27 '18 at 17:43
I don't know exactly which step you are referring to but you will need that at some point.
– timur
Dec 27 '18 at 19:50
|
show 2 more comments
We have
$$
|int fu| leq |f|_{L^2}|u|_{L^2}
leq
varepsilon|u|_{L^2}^2+frac4varepsilon|f|_{L^2}^2
leq
Cvarepsilon|Du|_{L^2}^2+frac4varepsilon|f|_{L^2}^2 ,
$$
where we have used the fact that $int u=0$ in combination with the Poincare-Wirtinger inequality. Take $varepsilon>0$ small enough, and get
$$
|Du|_{L^4}^4leqalpha J(u) + beta
$$
for some constants $alpha,beta$.
answered Dec 26 '18 at 20:55
timur
11.8k1943
Is it obvious that we can apply Poincare-Wirtinger in this way for all $u in W^{1,4}(mathbb{T}^2)$
– rogerroger
Dec 26 '18 at 21:05
Thanks by the way, I should of thought to apply the epsilon young inequality it is used everywhere :)
– rogerroger
Dec 26 '18 at 21:06
1
@rogerroger: Yes, because the domain is bounded, there is the continuous embedding $W^{1,4}subset W^{1,2}$.
– timur
Dec 26 '18 at 21:10
and I assume you have used $u$ having zero average by not regarding it in your $||cdot ||_{W^{1,4}}$ norm?
– rogerroger
Dec 27 '18 at 17:43
I don't know exactly which step you are referring to but you will need that at some point.
– timur
Dec 27 '18 at 19:50
|
show 2 more comments
We have
$$
|int fu| leq |f|_{L^2}|u|_{L^2}
leq
varepsilon|u|_{L^2}^2+frac4varepsilon|f|_{L^2}^2
leq
Cvarepsilon|Du|_{L^2}^2+frac4varepsilon|f|_{L^2}^2 ,
$$
where we have used the fact that $int u=0$ in combination with the Poincare-Wirtinger inequality. Take $varepsilon>0$ small enough, and get
$$
|Du|_{L^4}^4leqalpha J(u) + beta
$$
for some constants $alpha,beta$.
answered Dec 26 '18 at 20:55
timur
11.8k1943
We have
$$
|int fu| leq |f|_{L^2}|u|_{L^2}
leq
varepsilon|u|_{L^2}^2+frac4varepsilon|f|_{L^2}^2
leq
Cvarepsilon|Du|_{L^2}^2+frac4varepsilon|f|_{L^2}^2 ,
$$
where we have used the fact that $int u=0$ in combination with the Poincare-Wirtinger inequality. Take $varepsilon>0$ small enough, and get
$$
|Du|_{L^4}^4leqalpha J(u) + beta
$$
for some constants $alpha,beta$.
answered Dec 26 '18 at 20:55
timur
11.8k1943
answered Dec 26 '18 at 20:55
timur
11.8k1943
answered Dec 26 '18 at 20:55
timur
11.8k1943
answered Dec 26 '18 at 20:55
timur
11.8k1943
11.8k1943
Is it obvious that we can apply Poincare-Wirtinger in this way for all $u in W^{1,4}(mathbb{T}^2)$
– rogerroger
Dec 26 '18 at 21:05
Thanks by the way, I should of thought to apply the epsilon young inequality it is used everywhere :)
– rogerroger
Dec 26 '18 at 21:06
1
@rogerroger: Yes, because the domain is bounded, there is the continuous embedding $W^{1,4}subset W^{1,2}$.
– timur
Dec 26 '18 at 21:10
and I assume you have used $u$ having zero average by not regarding it in your $||cdot ||_{W^{1,4}}$ norm?
– rogerroger
Dec 27 '18 at 17:43
I don't know exactly which step you are referring to but you will need that at some point.
– timur
Dec 27 '18 at 19:50
|
show 2 more comments
Is it obvious that we can apply Poincare-Wirtinger in this way for all $u in W^{1,4}(mathbb{T}^2)$
– rogerroger
Dec 26 '18 at 21:05
Thanks by the way, I should of thought to apply the epsilon young inequality it is used everywhere :)
– rogerroger
Dec 26 '18 at 21:06
1
@rogerroger: Yes, because the domain is bounded, there is the continuous embedding $W^{1,4}subset W^{1,2}$.
– timur
Dec 26 '18 at 21:10
and I assume you have used $u$ having zero average by not regarding it in your $||cdot ||_{W^{1,4}}$ norm?
– rogerroger
Dec 27 '18 at 17:43
I don't know exactly which step you are referring to but you will need that at some point.
– timur
Dec 27 '18 at 19:50
Is it obvious that we can apply Poincare-Wirtinger in this way for all $u in W^{1,4}(mathbb{T}^2)$
– rogerroger
Dec 26 '18 at 21:05
Is it obvious that we can apply Poincare-Wirtinger in this way for all $u in W^{1,4}(mathbb{T}^2)$
– rogerroger
Dec 26 '18 at 21:05
Thanks by the way, I should of thought to apply the epsilon young inequality it is used everywhere :)
– rogerroger
Dec 26 '18 at 21:06
Thanks by the way, I should of thought to apply the epsilon young inequality it is used everywhere :)
– rogerroger
Dec 26 '18 at 21:06
1
1
@rogerroger: Yes, because the domain is bounded, there is the continuous embedding $W^{1,4}subset W^{1,2}$.
– timur
Dec 26 '18 at 21:10
@rogerroger: Yes, because the domain is bounded, there is the continuous embedding $W^{1,4}subset W^{1,2}$.
– timur
Dec 26 '18 at 21:10
and I assume you have used $u$ having zero average by not regarding it in your $||cdot ||_{W^{1,4}}$ norm?
– rogerroger
Dec 27 '18 at 17:43
and I assume you have used $u$ having zero average by not regarding it in your $||cdot ||_{W^{1,4}}$ norm?
– rogerroger
Dec 27 '18 at 17:43
I don't know exactly which step you are referring to but you will need that at some point.
– timur
Dec 27 '18 at 19:50
I don't know exactly which step you are referring to but you will need that at some point.
– timur
Dec 27 '18 at 19:50
|
show 2 more comments
rogerroger is a new contributor. Be nice, and check out our Code of Conduct.
rogerroger is a new contributor. Be nice, and check out our Code of Conduct.
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What is $Delta$?
– timur
Dec 26 '18 at 19:43
@timur Sorry I meant to write $||Du||_{W^{1,4}(mathbb{T}^2)}$, I will edit now
– rogerroger
Dec 26 '18 at 19:59