Continuity of Energy Functional












3












$begingroup$


Let $u : Omega times [0,T]$ be a function such that $u in C^{2,1}(Omega times [0,T])cap C^{1}((0,T);L^{2}(Omega))cap C([0,T);H_{0}^{1}(Omega))$ for $Omega subset mathbb{R}$ an unbounded domain.



In order to clarify the meaning of the notation, I will explain some notations mentioned above.

1. $C^{2,1}(Omega times [0,T])$ : the function is twice differentiable with respect to spatial domain and once differentiable with respect to time domain.

2. $C^{1}((0,T);L^{2}(Omega))$ : for any fixed $t in (0,T)$, $u(, ., ,t)in L^{2}(Omega)$ and the mapping is once differentiable

3. $C([0,T);H_{0}^{1}(Omega))$ : for any fixed $t in [0,T), u(, ., ,t)in H_{0}^{1}(Omega)$ and the mapping is continuous.



So now I define, a functional $F[, .,] = ||, .,||_{L^{p}(Omega)}^{p}$ ($2<p<infty$) so that I have $F[u(,.,)] : [0,T)tomathbb{R}$



What I want to show is $F[u(,.,)] in C([0,T);mathbb{R})$ but I am not sure how to ensure that $||,.,||_{L^{p}(Omega)}$ is finite since $Omega$ is unbounded and thus I cannot use the embedding of $L^{p}$ to $L^{2}$ for $2<p<infty$. Furthermore, the dimension $n=1$ so I cannot use any embedding inequality here.



Any help is much appreciated! Thank you!






functional-analysis functions continuity normed-spaces







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  • $begingroup$
    Halo Kak Evan :)
    $endgroup$
    – Sou
    Jan 18 at 4:07










  • $begingroup$
    Hello there, since this is an English Forum. I'd rather not to speak in Indonesian Language. I assume I know you from my undergraduate university?
    $endgroup$
    – Evan William Chandra
    Jan 18 at 4:11










  • $begingroup$
    Yes. Just saying hello.
    $endgroup$
    – Sou
    Jan 18 at 4:12
















3












$begingroup$


Let $u : Omega times [0,T]$ be a function such that $u in C^{2,1}(Omega times [0,T])cap C^{1}((0,T);L^{2}(Omega))cap C([0,T);H_{0}^{1}(Omega))$ for $Omega subset mathbb{R}$ an unbounded domain.



In order to clarify the meaning of the notation, I will explain some notations mentioned above.

1. $C^{2,1}(Omega times [0,T])$ : the function is twice differentiable with respect to spatial domain and once differentiable with respect to time domain.

2. $C^{1}((0,T);L^{2}(Omega))$ : for any fixed $t in (0,T)$, $u(, ., ,t)in L^{2}(Omega)$ and the mapping is once differentiable

3. $C([0,T);H_{0}^{1}(Omega))$ : for any fixed $t in [0,T), u(, ., ,t)in H_{0}^{1}(Omega)$ and the mapping is continuous.



So now I define, a functional $F[, .,] = ||, .,||_{L^{p}(Omega)}^{p}$ ($2<p<infty$) so that I have $F[u(,.,)] : [0,T)tomathbb{R}$



What I want to show is $F[u(,.,)] in C([0,T);mathbb{R})$ but I am not sure how to ensure that $||,.,||_{L^{p}(Omega)}$ is finite since $Omega$ is unbounded and thus I cannot use the embedding of $L^{p}$ to $L^{2}$ for $2<p<infty$. Furthermore, the dimension $n=1$ so I cannot use any embedding inequality here.



Any help is much appreciated! Thank you!






functional-analysis functions continuity normed-spaces







share|cite|improve this question









$endgroup$












  • $begingroup$
    Halo Kak Evan :)
    $endgroup$
    – Sou
    Jan 18 at 4:07










  • $begingroup$
    Hello there, since this is an English Forum. I'd rather not to speak in Indonesian Language. I assume I know you from my undergraduate university?
    $endgroup$
    – Evan William Chandra
    Jan 18 at 4:11










  • $begingroup$
    Yes. Just saying hello.
    $endgroup$
    – Sou
    Jan 18 at 4:12














3












3








3





$begingroup$


Let $u : Omega times [0,T]$ be a function such that $u in C^{2,1}(Omega times [0,T])cap C^{1}((0,T);L^{2}(Omega))cap C([0,T);H_{0}^{1}(Omega))$ for $Omega subset mathbb{R}$ an unbounded domain.



In order to clarify the meaning of the notation, I will explain some notations mentioned above.

1. $C^{2,1}(Omega times [0,T])$ : the function is twice differentiable with respect to spatial domain and once differentiable with respect to time domain.

2. $C^{1}((0,T);L^{2}(Omega))$ : for any fixed $t in (0,T)$, $u(, ., ,t)in L^{2}(Omega)$ and the mapping is once differentiable

3. $C([0,T);H_{0}^{1}(Omega))$ : for any fixed $t in [0,T), u(, ., ,t)in H_{0}^{1}(Omega)$ and the mapping is continuous.



So now I define, a functional $F[, .,] = ||, .,||_{L^{p}(Omega)}^{p}$ ($2<p<infty$) so that I have $F[u(,.,)] : [0,T)tomathbb{R}$



What I want to show is $F[u(,.,)] in C([0,T);mathbb{R})$ but I am not sure how to ensure that $||,.,||_{L^{p}(Omega)}$ is finite since $Omega$ is unbounded and thus I cannot use the embedding of $L^{p}$ to $L^{2}$ for $2<p<infty$. Furthermore, the dimension $n=1$ so I cannot use any embedding inequality here.



Any help is much appreciated! Thank you!






functional-analysis functions continuity normed-spaces







share|cite|improve this question









$endgroup$




Let $u : Omega times [0,T]$ be a function such that $u in C^{2,1}(Omega times [0,T])cap C^{1}((0,T);L^{2}(Omega))cap C([0,T);H_{0}^{1}(Omega))$ for $Omega subset mathbb{R}$ an unbounded domain.



In order to clarify the meaning of the notation, I will explain some notations mentioned above.

1. $C^{2,1}(Omega times [0,T])$ : the function is twice differentiable with respect to spatial domain and once differentiable with respect to time domain.

2. $C^{1}((0,T);L^{2}(Omega))$ : for any fixed $t in (0,T)$, $u(, ., ,t)in L^{2}(Omega)$ and the mapping is once differentiable

3. $C([0,T);H_{0}^{1}(Omega))$ : for any fixed $t in [0,T), u(, ., ,t)in H_{0}^{1}(Omega)$ and the mapping is continuous.



So now I define, a functional $F[, .,] = ||, .,||_{L^{p}(Omega)}^{p}$ ($2<p<infty$) so that I have $F[u(,.,)] : [0,T)tomathbb{R}$



What I want to show is $F[u(,.,)] in C([0,T);mathbb{R})$ but I am not sure how to ensure that $||,.,||_{L^{p}(Omega)}$ is finite since $Omega$ is unbounded and thus I cannot use the embedding of $L^{p}$ to $L^{2}$ for $2<p<infty$. Furthermore, the dimension $n=1$ so I cannot use any embedding inequality here.



Any help is much appreciated! Thank you!






functional-analysis functions continuity normed-spaces




functional-analysis functions continuity normed-spaces






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share|cite|improve this question











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asked Jan 18 at 3:35









Evan William ChandraEvan William Chandra

619313




619313












  • $begingroup$
    Halo Kak Evan :)
    $endgroup$
    – Sou
    Jan 18 at 4:07










  • $begingroup$
    Hello there, since this is an English Forum. I'd rather not to speak in Indonesian Language. I assume I know you from my undergraduate university?
    $endgroup$
    – Evan William Chandra
    Jan 18 at 4:11










  • $begingroup$
    Yes. Just saying hello.
    $endgroup$
    – Sou
    Jan 18 at 4:12


















  • $begingroup$
    Halo Kak Evan :)
    $endgroup$
    – Sou
    Jan 18 at 4:07










  • $begingroup$
    Hello there, since this is an English Forum. I'd rather not to speak in Indonesian Language. I assume I know you from my undergraduate university?
    $endgroup$
    – Evan William Chandra
    Jan 18 at 4:11










  • $begingroup$
    Yes. Just saying hello.
    $endgroup$
    – Sou
    Jan 18 at 4:12
















$begingroup$
Halo Kak Evan :)
$endgroup$
– Sou
Jan 18 at 4:07




$begingroup$
Halo Kak Evan :)
$endgroup$
– Sou
Jan 18 at 4:07












$begingroup$
Hello there, since this is an English Forum. I'd rather not to speak in Indonesian Language. I assume I know you from my undergraduate university?
$endgroup$
– Evan William Chandra
Jan 18 at 4:11




$begingroup$
Hello there, since this is an English Forum. I'd rather not to speak in Indonesian Language. I assume I know you from my undergraduate university?
$endgroup$
– Evan William Chandra
Jan 18 at 4:11












$begingroup$
Yes. Just saying hello.
$endgroup$
– Sou
Jan 18 at 4:12




$begingroup$
Yes. Just saying hello.
$endgroup$
– Sou
Jan 18 at 4:12










1 Answer
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$begingroup$

What you need is the Sobolev embedding $H^1_0(Omega)hookrightarrow L^infty(Omega)$ (see for example Theorem 8.8 in Functional Analysis, Sobolev Spaces and Partial Differential Equations by Brezis). It implies $uin C([0,T);L^2(Omega))cap C([0,T);L^infty(Omega))$. Together with the interpolation inequality
$$
|f|_p^pleq |f|_2^{2}|f|_infty^{p-2}
$$
this yields $uin C([0,T);L^p(Omega))$. In particular, $F$ is continuous on $[0,T)$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thank you very much for your help!
    $endgroup$
    – Evan William Chandra
    Jan 21 at 0:47











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1 Answer
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active

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1 Answer
1






active

oldest

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active

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1












$begingroup$

What you need is the Sobolev embedding $H^1_0(Omega)hookrightarrow L^infty(Omega)$ (see for example Theorem 8.8 in Functional Analysis, Sobolev Spaces and Partial Differential Equations by Brezis). It implies $uin C([0,T);L^2(Omega))cap C([0,T);L^infty(Omega))$. Together with the interpolation inequality
$$
|f|_p^pleq |f|_2^{2}|f|_infty^{p-2}
$$
this yields $uin C([0,T);L^p(Omega))$. In particular, $F$ is continuous on $[0,T)$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thank you very much for your help!
    $endgroup$
    – Evan William Chandra
    Jan 21 at 0:47
















1












$begingroup$

What you need is the Sobolev embedding $H^1_0(Omega)hookrightarrow L^infty(Omega)$ (see for example Theorem 8.8 in Functional Analysis, Sobolev Spaces and Partial Differential Equations by Brezis). It implies $uin C([0,T);L^2(Omega))cap C([0,T);L^infty(Omega))$. Together with the interpolation inequality
$$
|f|_p^pleq |f|_2^{2}|f|_infty^{p-2}
$$
this yields $uin C([0,T);L^p(Omega))$. In particular, $F$ is continuous on $[0,T)$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thank you very much for your help!
    $endgroup$
    – Evan William Chandra
    Jan 21 at 0:47














1












1








1





$begingroup$

What you need is the Sobolev embedding $H^1_0(Omega)hookrightarrow L^infty(Omega)$ (see for example Theorem 8.8 in Functional Analysis, Sobolev Spaces and Partial Differential Equations by Brezis). It implies $uin C([0,T);L^2(Omega))cap C([0,T);L^infty(Omega))$. Together with the interpolation inequality
$$
|f|_p^pleq |f|_2^{2}|f|_infty^{p-2}
$$
this yields $uin C([0,T);L^p(Omega))$. In particular, $F$ is continuous on $[0,T)$.






share|cite|improve this answer









$endgroup$



What you need is the Sobolev embedding $H^1_0(Omega)hookrightarrow L^infty(Omega)$ (see for example Theorem 8.8 in Functional Analysis, Sobolev Spaces and Partial Differential Equations by Brezis). It implies $uin C([0,T);L^2(Omega))cap C([0,T);L^infty(Omega))$. Together with the interpolation inequality
$$
|f|_p^pleq |f|_2^{2}|f|_infty^{p-2}
$$
this yields $uin C([0,T);L^p(Omega))$. In particular, $F$ is continuous on $[0,T)$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 18 at 12:42









MaoWaoMaoWao

3,153617




3,153617












  • $begingroup$
    Thank you very much for your help!
    $endgroup$
    – Evan William Chandra
    Jan 21 at 0:47


















  • $begingroup$
    Thank you very much for your help!
    $endgroup$
    – Evan William Chandra
    Jan 21 at 0:47
















$begingroup$
Thank you very much for your help!
$endgroup$
– Evan William Chandra
Jan 21 at 0:47




$begingroup$
Thank you very much for your help!
$endgroup$
– Evan William Chandra
Jan 21 at 0:47


















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