Bounded sequence in Sobolev Space












3














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'.










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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
















3














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'.










share|cite|improve this question









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.




















  • 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














3












3








3







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'.










share|cite|improve this question









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






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edited Dec 28 '18 at 23:03





















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asked Dec 26 '18 at 17:48









rogerroger

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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


















  • 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










1 Answer
1






active

oldest

votes


















1














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 $alpha,beta$.






share|cite|improve this answer





















  • 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











Your Answer





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1 Answer
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1 Answer
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1














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 $alpha,beta$.






share|cite|improve this answer





















  • 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
















1














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 $alpha,beta$.






share|cite|improve this answer





















  • 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














1












1








1






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 $alpha,beta$.






share|cite|improve this answer












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 $alpha,beta$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










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


















  • 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










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