Existence of smooth function which has compact support
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Assume we have two sets $V,W$, for which $V subset subset W subset subset Omega subset mathbb{R}^n$ holds. Now we want to find a smooth function $psi$, such that
begin{equation*}
begin{cases}
psi equiv 1 text{ in } V, \
psi equiv 0 text{ in } mathbb{R}^n setminus W, \
psi in [0,1].
end{cases}
end{equation*}
holds.
How do we know that such a function exists? That problem occured to me in a proof in Chapter 6 of L. Evans' Book "Partial differential equations".
real-analysis functions pde special-functions
asked Jan 16 at 8:31
MaxMax
586
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add a comment |
$begingroup$
Assume we have two sets $V,W$, for which $V subset subset W subset subset Omega subset mathbb{R}^n$ holds. Now we want to find a smooth function $psi$, such that
begin{equation*}
begin{cases}
psi equiv 1 text{ in } V, \
psi equiv 0 text{ in } mathbb{R}^n setminus W, \
psi in [0,1].
end{cases}
end{equation*}
holds.
How do we know that such a function exists? That problem occured to me in a proof in Chapter 6 of L. Evans' Book "Partial differential equations".
real-analysis functions pde special-functions
asked Jan 16 at 8:31
MaxMax
586
$endgroup$
2
$begingroup$
Search for 'bump functions'. You can look at mathworld.wolfram.com/BumpFunction.html
$endgroup$
– Kavi Rama Murthy
Jan 16 at 8:41
4
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This follows from Urysohn Lemma. Your sets $V$ and $mathbb{R}^n-W$ are separated, since $V$ is compactly contained in the open set $W$ so there is room for such a function to exist. You can see the proof, for ex. on Wikipedia.
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– GReyes
Jan 16 at 8:41
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What does $subset subset $ mean? And what is $Omega $ ?
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– DanielWainfleet
Jan 16 at 23:18
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@DanielWainfleet it usually mean
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– AD.
Jan 17 at 0:22
$begingroup$
$subset subset$ means a compact subset. Thanks @GReyes, that fully answered my question! :)
$endgroup$
– Max
Jan 17 at 10:56
add a comment |
$begingroup$
Assume we have two sets $V,W$, for which $V subset subset W subset subset Omega subset mathbb{R}^n$ holds. Now we want to find a smooth function $psi$, such that
begin{equation*}
begin{cases}
psi equiv 1 text{ in } V, \
psi equiv 0 text{ in } mathbb{R}^n setminus W, \
psi in [0,1].
end{cases}
end{equation*}
holds.
How do we know that such a function exists? That problem occured to me in a proof in Chapter 6 of L. Evans' Book "Partial differential equations".
real-analysis functions pde special-functions
asked Jan 16 at 8:31
MaxMax
586
$endgroup$
Assume we have two sets $V,W$, for which $V subset subset W subset subset Omega subset mathbb{R}^n$ holds. Now we want to find a smooth function $psi$, such that
begin{equation*}
begin{cases}
psi equiv 1 text{ in } V, \
psi equiv 0 text{ in } mathbb{R}^n setminus W, \
psi in [0,1].
end{cases}
end{equation*}
holds.
How do we know that such a function exists? That problem occured to me in a proof in Chapter 6 of L. Evans' Book "Partial differential equations".
real-analysis functions pde special-functions
real-analysis functions pde special-functions
asked Jan 16 at 8:31
MaxMax
586
asked Jan 16 at 8:31
MaxMax
586
asked Jan 16 at 8:31
MaxMax
586
asked Jan 16 at 8:31
MaxMax
586
asked Jan 16 at 8:31
MaxMax
586
586
2
$begingroup$
Search for 'bump functions'. You can look at mathworld.wolfram.com/BumpFunction.html
$endgroup$
– Kavi Rama Murthy
Jan 16 at 8:41
4
$begingroup$
This follows from Urysohn Lemma. Your sets $V$ and $mathbb{R}^n-W$ are separated, since $V$ is compactly contained in the open set $W$ so there is room for such a function to exist. You can see the proof, for ex. on Wikipedia.
$endgroup$
– GReyes
Jan 16 at 8:41
$begingroup$
What does $subset subset $ mean? And what is $Omega $ ?
$endgroup$
– DanielWainfleet
Jan 16 at 23:18
$begingroup$
@DanielWainfleet it usually mean
$endgroup$
– AD.
Jan 17 at 0:22
$begingroup$
$subset subset$ means a compact subset. Thanks @GReyes, that fully answered my question! :)
$endgroup$
– Max
Jan 17 at 10:56
add a comment |
2
$begingroup$
Search for 'bump functions'. You can look at mathworld.wolfram.com/BumpFunction.html
$endgroup$
– Kavi Rama Murthy
Jan 16 at 8:41
4
$begingroup$
This follows from Urysohn Lemma. Your sets $V$ and $mathbb{R}^n-W$ are separated, since $V$ is compactly contained in the open set $W$ so there is room for such a function to exist. You can see the proof, for ex. on Wikipedia.
$endgroup$
– GReyes
Jan 16 at 8:41
$begingroup$
What does $subset subset $ mean? And what is $Omega $ ?
$endgroup$
– DanielWainfleet
Jan 16 at 23:18
$begingroup$
@DanielWainfleet it usually mean
$endgroup$
– AD.
Jan 17 at 0:22
$begingroup$
$subset subset$ means a compact subset. Thanks @GReyes, that fully answered my question! :)
$endgroup$
– Max
Jan 17 at 10:56
2
2
$begingroup$
Search for 'bump functions'. You can look at mathworld.wolfram.com/BumpFunction.html
$endgroup$
– Kavi Rama Murthy
Jan 16 at 8:41
$begingroup$
Search for 'bump functions'. You can look at mathworld.wolfram.com/BumpFunction.html
$endgroup$
– Kavi Rama Murthy
Jan 16 at 8:41
4
4
$begingroup$
This follows from Urysohn Lemma. Your sets $V$ and $mathbb{R}^n-W$ are separated, since $V$ is compactly contained in the open set $W$ so there is room for such a function to exist. You can see the proof, for ex. on Wikipedia.
$endgroup$
– GReyes
Jan 16 at 8:41
$begingroup$
This follows from Urysohn Lemma. Your sets $V$ and $mathbb{R}^n-W$ are separated, since $V$ is compactly contained in the open set $W$ so there is room for such a function to exist. You can see the proof, for ex. on Wikipedia.
$endgroup$
– GReyes
Jan 16 at 8:41
$begingroup$
What does $subset subset $ mean? And what is $Omega $ ?
$endgroup$
– DanielWainfleet
Jan 16 at 23:18
$begingroup$
What does $subset subset $ mean? And what is $Omega $ ?
$endgroup$
– DanielWainfleet
Jan 16 at 23:18
$begingroup$
@DanielWainfleet it usually mean
$endgroup$
– AD.
Jan 17 at 0:22
$begingroup$
@DanielWainfleet it usually mean
$endgroup$
– AD.
Jan 17 at 0:22
$begingroup$
$subset subset$ means a compact subset. Thanks @GReyes, that fully answered my question! :)
$endgroup$
– Max
Jan 17 at 10:56
$begingroup$
$subset subset$ means a compact subset. Thanks @GReyes, that fully answered my question! :)
$endgroup$
– Max
Jan 17 at 10:56
add a comment |
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2
$begingroup$
Search for 'bump functions'. You can look at mathworld.wolfram.com/BumpFunction.html
$endgroup$
– Kavi Rama Murthy
Jan 16 at 8:41
4
$begingroup$
This follows from Urysohn Lemma. Your sets $V$ and $mathbb{R}^n-W$ are separated, since $V$ is compactly contained in the open set $W$ so there is room for such a function to exist. You can see the proof, for ex. on Wikipedia.
$endgroup$
– GReyes
Jan 16 at 8:41
$begingroup$
What does $subset subset $ mean? And what is $Omega $ ?
$endgroup$
– DanielWainfleet
Jan 16 at 23:18
$begingroup$
@DanielWainfleet it usually mean
$endgroup$
– AD.
Jan 17 at 0:22
$begingroup$
$subset subset$ means a compact subset. Thanks @GReyes, that fully answered my question! :)
$endgroup$
– Max
Jan 17 at 10:56