Why is the distributional derivative of a continuous function non-atomic?
$begingroup$
For $Omega subset mathbb{R}^N$ and $u in BV(Omega)$, we define the distributional derivative $Du$ as the N-tuple of finite Radon measures in $Omega$, $Du = (D_1u,...,D_Nu)$, such that
$$
int_{Omega} u frac{partial phi}{partial x_i} mathrm{d}x = - int_{Omega} phi D_i uquadforall phi in C_c^{infty}(Omega),;i=1,ldots,N
$$
The question is in the title, thanks in advance!
measure-theory calculus-of-variations bounded-variation
edited Jan 20 at 20:04
Daniele Tampieri
2,65221022
asked Jan 9 at 18:48
Hari.M.S.Hari.M.S.
358
$endgroup$
add a comment |
$begingroup$
For $Omega subset mathbb{R}^N$ and $u in BV(Omega)$, we define the distributional derivative $Du$ as the N-tuple of finite Radon measures in $Omega$, $Du = (D_1u,...,D_Nu)$, such that
$$
int_{Omega} u frac{partial phi}{partial x_i} mathrm{d}x = - int_{Omega} phi D_i uquadforall phi in C_c^{infty}(Omega),;i=1,ldots,N
$$
The question is in the title, thanks in advance!
measure-theory calculus-of-variations bounded-variation
edited Jan 20 at 20:04
Daniele Tampieri
2,65221022
asked Jan 9 at 18:48
Hari.M.S.Hari.M.S.
358
$endgroup$
$begingroup$
$Du$ is a finite Radon measure $iff uin BV(Omega)$: I corrected your post accordingly.
$endgroup$
– Daniele Tampieri
Jan 20 at 20:05
add a comment |
$begingroup$
For $Omega subset mathbb{R}^N$ and $u in BV(Omega)$, we define the distributional derivative $Du$ as the N-tuple of finite Radon measures in $Omega$, $Du = (D_1u,...,D_Nu)$, such that
$$
int_{Omega} u frac{partial phi}{partial x_i} mathrm{d}x = - int_{Omega} phi D_i uquadforall phi in C_c^{infty}(Omega),;i=1,ldots,N
$$
The question is in the title, thanks in advance!
measure-theory calculus-of-variations bounded-variation
edited Jan 20 at 20:04
Daniele Tampieri
2,65221022
asked Jan 9 at 18:48
Hari.M.S.Hari.M.S.
358
$endgroup$
For $Omega subset mathbb{R}^N$ and $u in BV(Omega)$, we define the distributional derivative $Du$ as the N-tuple of finite Radon measures in $Omega$, $Du = (D_1u,...,D_Nu)$, such that
$$
int_{Omega} u frac{partial phi}{partial x_i} mathrm{d}x = - int_{Omega} phi D_i uquadforall phi in C_c^{infty}(Omega),;i=1,ldots,N
$$
The question is in the title, thanks in advance!
measure-theory calculus-of-variations bounded-variation
measure-theory calculus-of-variations bounded-variation
edited Jan 20 at 20:04
Daniele Tampieri
2,65221022
asked Jan 9 at 18:48
Hari.M.S.Hari.M.S.
358
edited Jan 20 at 20:04
Daniele Tampieri
2,65221022
asked Jan 9 at 18:48
Hari.M.S.Hari.M.S.
358
edited Jan 20 at 20:04
Daniele Tampieri
2,65221022
edited Jan 20 at 20:04
Daniele Tampieri
2,65221022
edited Jan 20 at 20:04
Daniele Tampieri
2,65221022
2,65221022
asked Jan 9 at 18:48
Hari.M.S.Hari.M.S.
358
asked Jan 9 at 18:48
Hari.M.S.Hari.M.S.
358
asked Jan 9 at 18:48
Hari.M.S.Hari.M.S.
358
358
$begingroup$
$Du$ is a finite Radon measure $iff uin BV(Omega)$: I corrected your post accordingly.
$endgroup$
– Daniele Tampieri
Jan 20 at 20:05
add a comment |
$begingroup$
$Du$ is a finite Radon measure $iff uin BV(Omega)$: I corrected your post accordingly.
$endgroup$
– Daniele Tampieri
Jan 20 at 20:05
$begingroup$
$Du$ is a finite Radon measure $iff uin BV(Omega)$: I corrected your post accordingly.
$endgroup$
– Daniele Tampieri
Jan 20 at 20:05
$begingroup$
$Du$ is a finite Radon measure $iff uin BV(Omega)$: I corrected your post accordingly.
$endgroup$
– Daniele Tampieri
Jan 20 at 20:05
add a comment |
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$begingroup$
$Du$ is a finite Radon measure $iff uin BV(Omega)$: I corrected your post accordingly.
$endgroup$
– Daniele Tampieri
Jan 20 at 20:05