Lower bound of the total variation
$begingroup$
Let $fin C^1(Omega)$ be a continuously differentiable function on a domain $Omega subset mathbb{R}^n$. It is known that the total variation of $f$ is equal to
$$V(f,Omega)=int_Omega |nabla f(x)| text{d}x$$
In one dimension $n=1$ and $Omega=[a,b]$ it is not hard to show that
$$V(f,[a,b])=int_a^b |f'(x)| text{d}xgeq max_{[a,b]} f - min_{[a,b]} f$$
Is there a similar bound that can be obtained in greater dimensions $(n>1)$?
real-analysis multivariable-calculus total-variation bounds-of-integration
asked Jan 25 at 23:22
MOMOMOMO
717312
$endgroup$
add a comment |
$begingroup$
Let $fin C^1(Omega)$ be a continuously differentiable function on a domain $Omega subset mathbb{R}^n$. It is known that the total variation of $f$ is equal to
$$V(f,Omega)=int_Omega |nabla f(x)| text{d}x$$
In one dimension $n=1$ and $Omega=[a,b]$ it is not hard to show that
$$V(f,[a,b])=int_a^b |f'(x)| text{d}xgeq max_{[a,b]} f - min_{[a,b]} f$$
Is there a similar bound that can be obtained in greater dimensions $(n>1)$?
real-analysis multivariable-calculus total-variation bounds-of-integration
asked Jan 25 at 23:22
MOMOMOMO
717312
$endgroup$
add a comment |
$begingroup$
Let $fin C^1(Omega)$ be a continuously differentiable function on a domain $Omega subset mathbb{R}^n$. It is known that the total variation of $f$ is equal to
$$V(f,Omega)=int_Omega |nabla f(x)| text{d}x$$
In one dimension $n=1$ and $Omega=[a,b]$ it is not hard to show that
$$V(f,[a,b])=int_a^b |f'(x)| text{d}xgeq max_{[a,b]} f - min_{[a,b]} f$$
Is there a similar bound that can be obtained in greater dimensions $(n>1)$?
real-analysis multivariable-calculus total-variation bounds-of-integration
asked Jan 25 at 23:22
MOMOMOMO
717312
$endgroup$
Let $fin C^1(Omega)$ be a continuously differentiable function on a domain $Omega subset mathbb{R}^n$. It is known that the total variation of $f$ is equal to
$$V(f,Omega)=int_Omega |nabla f(x)| text{d}x$$
In one dimension $n=1$ and $Omega=[a,b]$ it is not hard to show that
$$V(f,[a,b])=int_a^b |f'(x)| text{d}xgeq max_{[a,b]} f - min_{[a,b]} f$$
Is there a similar bound that can be obtained in greater dimensions $(n>1)$?
real-analysis multivariable-calculus total-variation bounds-of-integration
real-analysis multivariable-calculus total-variation bounds-of-integration
asked Jan 25 at 23:22
MOMOMOMO
717312
asked Jan 25 at 23:22
MOMOMOMO
717312
edited Jan 25 at 23:44
MOMO
asked Jan 25 at 23:22
MOMOMOMO
717312
asked Jan 25 at 23:22
MOMOMOMO
717312
asked Jan 25 at 23:22
MOMOMOMO
717312
717312
add a comment |
add a comment |
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