Solution to weighted $p$-Laplace equation
$begingroup$
Let $Omega$ be a bounded smooth domain in $mathbb{R}^N$ where $Ngeq 2$ and let $fin L^{infty}(Omega)$.
Then does there exist $win A_p$ (the class of Muckenhoupt weights) such that no solutions $u$ in the weighted Sobolev space $W_{0}^{1,p}(Omega,w)$ of the equation
$$
-text{div}(w(x)|nabla u|^{p-2}nabla u)=f,text{in},Omega,
$$
is continuous?
As far as I know, in case of $p=2$ and $f=0$, if one choose $win A_2$, then all the solutions are continuous follows from the result of Fabes-Serapioni.
Therefore, it is better to choose $fneq 0$ and construct such $w$. Even an example for $p=2$ case will be enough.
If possible, kindly help me.
Thanks in advance.
sobolev-spaces harmonic-analysis regularity-theory-of-pdes
asked Dec 23 '18 at 18:09
MathloverMathlover
1558
$endgroup$
add a comment |
$begingroup$
Let $Omega$ be a bounded smooth domain in $mathbb{R}^N$ where $Ngeq 2$ and let $fin L^{infty}(Omega)$.
Then does there exist $win A_p$ (the class of Muckenhoupt weights) such that no solutions $u$ in the weighted Sobolev space $W_{0}^{1,p}(Omega,w)$ of the equation
$$
-text{div}(w(x)|nabla u|^{p-2}nabla u)=f,text{in},Omega,
$$
is continuous?
As far as I know, in case of $p=2$ and $f=0$, if one choose $win A_2$, then all the solutions are continuous follows from the result of Fabes-Serapioni.
Therefore, it is better to choose $fneq 0$ and construct such $w$. Even an example for $p=2$ case will be enough.
If possible, kindly help me.
Thanks in advance.
sobolev-spaces harmonic-analysis regularity-theory-of-pdes
asked Dec 23 '18 at 18:09
MathloverMathlover
1558
$endgroup$
add a comment |
$begingroup$
Let $Omega$ be a bounded smooth domain in $mathbb{R}^N$ where $Ngeq 2$ and let $fin L^{infty}(Omega)$.
Then does there exist $win A_p$ (the class of Muckenhoupt weights) such that no solutions $u$ in the weighted Sobolev space $W_{0}^{1,p}(Omega,w)$ of the equation
$$
-text{div}(w(x)|nabla u|^{p-2}nabla u)=f,text{in},Omega,
$$
is continuous?
As far as I know, in case of $p=2$ and $f=0$, if one choose $win A_2$, then all the solutions are continuous follows from the result of Fabes-Serapioni.
Therefore, it is better to choose $fneq 0$ and construct such $w$. Even an example for $p=2$ case will be enough.
If possible, kindly help me.
Thanks in advance.
sobolev-spaces harmonic-analysis regularity-theory-of-pdes
asked Dec 23 '18 at 18:09
MathloverMathlover
1558
$endgroup$
Let $Omega$ be a bounded smooth domain in $mathbb{R}^N$ where $Ngeq 2$ and let $fin L^{infty}(Omega)$.
Then does there exist $win A_p$ (the class of Muckenhoupt weights) such that no solutions $u$ in the weighted Sobolev space $W_{0}^{1,p}(Omega,w)$ of the equation
$$
-text{div}(w(x)|nabla u|^{p-2}nabla u)=f,text{in},Omega,
$$
is continuous?
As far as I know, in case of $p=2$ and $f=0$, if one choose $win A_2$, then all the solutions are continuous follows from the result of Fabes-Serapioni.
Therefore, it is better to choose $fneq 0$ and construct such $w$. Even an example for $p=2$ case will be enough.
If possible, kindly help me.
Thanks in advance.
sobolev-spaces harmonic-analysis regularity-theory-of-pdes
sobolev-spaces harmonic-analysis regularity-theory-of-pdes
asked Dec 23 '18 at 18:09
MathloverMathlover
1558
asked Dec 23 '18 at 18:09
MathloverMathlover
1558
asked Dec 23 '18 at 18:09
MathloverMathlover
1558
asked Dec 23 '18 at 18:09
MathloverMathlover
1558
asked Dec 23 '18 at 18:09
MathloverMathlover
1558
1558
add a comment |
add a comment |
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