Demicontinuity implies continuity
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I want to check the continuity of the operator $A:Xto X^{*}$ defined by $$langle A(u),vrangle=int_{Omega}a(x,nabla u)cdotnabla v,dx-int_{Omega}g_k(u)v,dxquadtext{ for every }u,vin X.$$
Here $Omega$ is a smooth bounded domain in $mathbb{R}^N$ with $Ngeq 2$, $X=W_0^{1,p}(Omega)$ ($p>1$) and $X^{*}$ is the dual space of $X$.
Moreover, $a(x,s)$ is a Caratheodory function defined on $Omegatimesmathbb{R}^N$ to $mathbb{R}^N$, and $g_k(s)=text{min}{s^{-delta},k}$ for $k>0$ and $s>0$ and $g_k(s)=k$ for $sleq 0$. One can take $A(x,zeta)=|zeta|^{p-2}zeta$ for example.
To proceed, we need to prove that for every $u_nto u$ in $X$,
$$A(u_n)to A(u);text{ in }X^{*}.$$
I am able to prove that for every $u_nto u$ in $X$, one has $$langle A(u_n),vrangleto langle A(u),vranglequadtext{ for every }vin X.$$ In fact, such a property is called demicontinuity of $A$.
Now from here can I say $A(u_n)to A(u)$ in $X^{*}$, which will give the continuity of $A$?
Please help me.
Thanks in advance.
functional-analysis analysis pde sobolev-spaces
asked 2 days ago
Mathlover
988
add a comment |
up vote
1
down vote
favorite
I want to check the continuity of the operator $A:Xto X^{*}$ defined by $$langle A(u),vrangle=int_{Omega}a(x,nabla u)cdotnabla v,dx-int_{Omega}g_k(u)v,dxquadtext{ for every }u,vin X.$$
Here $Omega$ is a smooth bounded domain in $mathbb{R}^N$ with $Ngeq 2$, $X=W_0^{1,p}(Omega)$ ($p>1$) and $X^{*}$ is the dual space of $X$.
Moreover, $a(x,s)$ is a Caratheodory function defined on $Omegatimesmathbb{R}^N$ to $mathbb{R}^N$, and $g_k(s)=text{min}{s^{-delta},k}$ for $k>0$ and $s>0$ and $g_k(s)=k$ for $sleq 0$. One can take $A(x,zeta)=|zeta|^{p-2}zeta$ for example.
To proceed, we need to prove that for every $u_nto u$ in $X$,
$$A(u_n)to A(u);text{ in }X^{*}.$$
I am able to prove that for every $u_nto u$ in $X$, one has $$langle A(u_n),vrangleto langle A(u),vranglequadtext{ for every }vin X.$$ In fact, such a property is called demicontinuity of $A$.
Now from here can I say $A(u_n)to A(u)$ in $X^{*}$, which will give the continuity of $A$?
Please help me.
Thanks in advance.
functional-analysis analysis pde sobolev-spaces
asked 2 days ago
Mathlover
988
Note that you have shown that $A(u_n) rightharpoonup A(u)$ weakly in $X^*$ and you want to have $A(u_n) to A(u)$ strongly in $X^*$. It is clear that the latter is much stronger. Btw: Do you have any assumptions on $a$?
– gerw
yesterday
Yes, you may assume $a(x,zeta)=|zeta|^{p-2}zeta$. Thank you for pointing out this assumption.
– Mathlover
yesterday
I have edited the question, moreover $g_k$ has defined for any $sinmathbb{R}$.
– Mathlover
yesterday
Can you kindly have a look at the following question: math.stackexchange.com/questions/2996709/…
– Mathlover
18 hours ago
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I want to check the continuity of the operator $A:Xto X^{*}$ defined by $$langle A(u),vrangle=int_{Omega}a(x,nabla u)cdotnabla v,dx-int_{Omega}g_k(u)v,dxquadtext{ for every }u,vin X.$$
Here $Omega$ is a smooth bounded domain in $mathbb{R}^N$ with $Ngeq 2$, $X=W_0^{1,p}(Omega)$ ($p>1$) and $X^{*}$ is the dual space of $X$.
Moreover, $a(x,s)$ is a Caratheodory function defined on $Omegatimesmathbb{R}^N$ to $mathbb{R}^N$, and $g_k(s)=text{min}{s^{-delta},k}$ for $k>0$ and $s>0$ and $g_k(s)=k$ for $sleq 0$. One can take $A(x,zeta)=|zeta|^{p-2}zeta$ for example.
To proceed, we need to prove that for every $u_nto u$ in $X$,
$$A(u_n)to A(u);text{ in }X^{*}.$$
I am able to prove that for every $u_nto u$ in $X$, one has $$langle A(u_n),vrangleto langle A(u),vranglequadtext{ for every }vin X.$$ In fact, such a property is called demicontinuity of $A$.
Now from here can I say $A(u_n)to A(u)$ in $X^{*}$, which will give the continuity of $A$?
Please help me.
Thanks in advance.
functional-analysis analysis pde sobolev-spaces
asked 2 days ago
Mathlover
988
I want to check the continuity of the operator $A:Xto X^{*}$ defined by $$langle A(u),vrangle=int_{Omega}a(x,nabla u)cdotnabla v,dx-int_{Omega}g_k(u)v,dxquadtext{ for every }u,vin X.$$
Here $Omega$ is a smooth bounded domain in $mathbb{R}^N$ with $Ngeq 2$, $X=W_0^{1,p}(Omega)$ ($p>1$) and $X^{*}$ is the dual space of $X$.
Moreover, $a(x,s)$ is a Caratheodory function defined on $Omegatimesmathbb{R}^N$ to $mathbb{R}^N$, and $g_k(s)=text{min}{s^{-delta},k}$ for $k>0$ and $s>0$ and $g_k(s)=k$ for $sleq 0$. One can take $A(x,zeta)=|zeta|^{p-2}zeta$ for example.
To proceed, we need to prove that for every $u_nto u$ in $X$,
$$A(u_n)to A(u);text{ in }X^{*}.$$
I am able to prove that for every $u_nto u$ in $X$, one has $$langle A(u_n),vrangleto langle A(u),vranglequadtext{ for every }vin X.$$ In fact, such a property is called demicontinuity of $A$.
Now from here can I say $A(u_n)to A(u)$ in $X^{*}$, which will give the continuity of $A$?
Please help me.
Thanks in advance.
functional-analysis analysis pde sobolev-spaces
functional-analysis analysis pde sobolev-spaces
asked 2 days ago
Mathlover
988
asked 2 days ago
Mathlover
988
edited yesterday
asked 2 days ago
Mathlover
988
asked 2 days ago
Mathlover
988
asked 2 days ago
Mathlover
988
988
Note that you have shown that $A(u_n) rightharpoonup A(u)$ weakly in $X^*$ and you want to have $A(u_n) to A(u)$ strongly in $X^*$. It is clear that the latter is much stronger. Btw: Do you have any assumptions on $a$?
– gerw
yesterday
Yes, you may assume $a(x,zeta)=|zeta|^{p-2}zeta$. Thank you for pointing out this assumption.
– Mathlover
yesterday
I have edited the question, moreover $g_k$ has defined for any $sinmathbb{R}$.
– Mathlover
yesterday
Can you kindly have a look at the following question: math.stackexchange.com/questions/2996709/…
– Mathlover
18 hours ago
add a comment |
Note that you have shown that $A(u_n) rightharpoonup A(u)$ weakly in $X^*$ and you want to have $A(u_n) to A(u)$ strongly in $X^*$. It is clear that the latter is much stronger. Btw: Do you have any assumptions on $a$?
– gerw
yesterday
Yes, you may assume $a(x,zeta)=|zeta|^{p-2}zeta$. Thank you for pointing out this assumption.
– Mathlover
yesterday
I have edited the question, moreover $g_k$ has defined for any $sinmathbb{R}$.
– Mathlover
yesterday
Can you kindly have a look at the following question: math.stackexchange.com/questions/2996709/…
– Mathlover
18 hours ago
Note that you have shown that $A(u_n) rightharpoonup A(u)$ weakly in $X^*$ and you want to have $A(u_n) to A(u)$ strongly in $X^*$. It is clear that the latter is much stronger. Btw: Do you have any assumptions on $a$?
– gerw
yesterday
Note that you have shown that $A(u_n) rightharpoonup A(u)$ weakly in $X^*$ and you want to have $A(u_n) to A(u)$ strongly in $X^*$. It is clear that the latter is much stronger. Btw: Do you have any assumptions on $a$?
– gerw
yesterday
Yes, you may assume $a(x,zeta)=|zeta|^{p-2}zeta$. Thank you for pointing out this assumption.
– Mathlover
yesterday
Yes, you may assume $a(x,zeta)=|zeta|^{p-2}zeta$. Thank you for pointing out this assumption.
– Mathlover
yesterday
I have edited the question, moreover $g_k$ has defined for any $sinmathbb{R}$.
– Mathlover
yesterday
I have edited the question, moreover $g_k$ has defined for any $sinmathbb{R}$.
– Mathlover
yesterday
Can you kindly have a look at the following question: math.stackexchange.com/questions/2996709/…
– Mathlover
18 hours ago
Can you kindly have a look at the following question: math.stackexchange.com/questions/2996709/…
– Mathlover
18 hours ago
add a comment |
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Note that you have shown that $A(u_n) rightharpoonup A(u)$ weakly in $X^*$ and you want to have $A(u_n) to A(u)$ strongly in $X^*$. It is clear that the latter is much stronger. Btw: Do you have any assumptions on $a$?
– gerw
yesterday
Yes, you may assume $a(x,zeta)=|zeta|^{p-2}zeta$. Thank you for pointing out this assumption.
– Mathlover
yesterday
I have edited the question, moreover $g_k$ has defined for any $sinmathbb{R}$.
– Mathlover
yesterday
Can you kindly have a look at the following question: math.stackexchange.com/questions/2996709/…
– Mathlover
18 hours ago