Extended Global Approximation Theorem
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 approximation-theory
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 approximation-theory
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 approximation-theory
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 approximation-theory
analysis pde sobolev-spaces approximation-theory
edited 22 hours ago
asked 2 days ago
w.sdka
30419
30419
add a comment |
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2998025%2fextended-global-approximation-theorem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown