Extended Global Approximation Theorem in Sobolev Space
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In Evans,
$textbf{Theorem} $ (Global Approximation Theorem) Assume $U$ is bounded, and $partial U$ is $C^1$. Suppose as well that $u in W^{k,p}(U)$ for some $1leq p < infty$. Then, there exist functions $u_m in C^{infty}(bar{U})$ such that
begin{align*}
u_m rightarrow u quad textrm{ in } W^{k,p}(U)
end{align*}
$textbf{Question}$ Although we change the boundary condition like
begin{align*}
partial U=bigcup_{j=1}^n Gamma_j, quad (textrm{boundary is piecewise } C^{1})
end{align*}
where each $Gamma_j$ for $j=1, cdots, n$ is a $C^1$, $Gamma_j$ and $Gamma_{j^{'}}$ do not intersect except at their endpoints if $jneq j'$, then does the theorem still hold?
Any help is appreciated!!
I want to know references related that...
Thank you!!
analysis pde sobolev-spaces
asked yesterday
w.sdka
30219
add a comment |
up vote
1
down vote
favorite
In Evans,
$textbf{Theorem} $ (Global Approximation Theorem) Assume $U$ is bounded, and $partial U$ is $C^1$. Suppose as well that $u in W^{k,p}(U)$ for some $1leq p < infty$. Then, there exist functions $u_m in C^{infty}(bar{U})$ such that
begin{align*}
u_m rightarrow u quad textrm{ in } W^{k,p}(U)
end{align*}
$textbf{Question}$ Although we change the boundary condition like
begin{align*}
partial U=bigcup_{j=1}^n Gamma_j, quad (textrm{boundary is piecewise } C^{1})
end{align*}
where each $Gamma_j$ for $j=1, cdots, n$ is a $C^1$, $Gamma_j$ and $Gamma_{j^{'}}$ do not intersect except at their endpoints if $jneq j'$, then does the theorem still hold?
Any help is appreciated!!
I want to know references related that...
Thank you!!
analysis pde sobolev-spaces
asked yesterday
w.sdka
30219
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
In Evans,
$textbf{Theorem} $ (Global Approximation Theorem) Assume $U$ is bounded, and $partial U$ is $C^1$. Suppose as well that $u in W^{k,p}(U)$ for some $1leq p < infty$. Then, there exist functions $u_m in C^{infty}(bar{U})$ such that
begin{align*}
u_m rightarrow u quad textrm{ in } W^{k,p}(U)
end{align*}
$textbf{Question}$ Although we change the boundary condition like
begin{align*}
partial U=bigcup_{j=1}^n Gamma_j, quad (textrm{boundary is piecewise } C^{1})
end{align*}
where each $Gamma_j$ for $j=1, cdots, n$ is a $C^1$, $Gamma_j$ and $Gamma_{j^{'}}$ do not intersect except at their endpoints if $jneq j'$, then does the theorem still hold?
Any help is appreciated!!
I want to know references related that...
Thank you!!
analysis pde sobolev-spaces
asked yesterday
w.sdka
30219
In Evans,
$textbf{Theorem} $ (Global Approximation Theorem) Assume $U$ is bounded, and $partial U$ is $C^1$. Suppose as well that $u in W^{k,p}(U)$ for some $1leq p < infty$. Then, there exist functions $u_m in C^{infty}(bar{U})$ such that
begin{align*}
u_m rightarrow u quad textrm{ in } W^{k,p}(U)
end{align*}
$textbf{Question}$ Although we change the boundary condition like
begin{align*}
partial U=bigcup_{j=1}^n Gamma_j, quad (textrm{boundary is piecewise } C^{1})
end{align*}
where each $Gamma_j$ for $j=1, cdots, n$ is a $C^1$, $Gamma_j$ and $Gamma_{j^{'}}$ do not intersect except at their endpoints if $jneq j'$, then does the theorem still hold?
Any help is appreciated!!
I want to know references related that...
Thank you!!
analysis pde sobolev-spaces
analysis pde sobolev-spaces
asked yesterday
w.sdka
30219
asked yesterday
w.sdka
30219
edited 6 hours ago
asked yesterday
w.sdka
30219
asked yesterday
w.sdka
30219
asked yesterday
w.sdka
30219
30219
add a comment |
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