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Prove that if $E subset mathbb{R}^n $ has finite perimeter, then almost every vertical slice has finite...
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Before explaining my problem, I recall the definitions:
Let $E subset mathbb{R}^n$ be a Lebesgue measurable set. We say that $E$ is a set of locally finite perimeter if for every compact set $K subset mathbb{R}^n$ it holds that
begin{equation*} label{eq:deflocfiniteperimeter}
M_K := sup left{ int_{E} div, T(x) ,dx : T in C^1_c( mathbb{R}^n; mathbb{R}^n), spt ,T subset K, | T | leq 1 right} < infty.
end{equation*}
Moreover, if $$
supleft{M_K:K subset mathbb{R}^n, Ktext{compact}right}< infty,$$
then we say that $E$ is a set of finite perimeter.
Now suppose that $E$ is a set of finite perimeter. I have to prove that for $mathcal{L}^{n-1}$-a.e. $z in mathbb{R}^{n-1}$ the vertical slice $E_z subset mathbb{R}$ is a set of finite perimeter; where, by definition, $E_z := { t in mathbb{R} : (z,t ) in E}$.
I can only prove that for a.e. $z in mathbb{R}^{n-1}$ the vertical slice is a set of LOCALLY finite perimeter. Here I descrive my attempt:
Let $rho$ is a regulatizing kernel and consider the sequence $ u_{h} : = chi_E * rho_{1/h}$: I already know (from a previous result) that
$$ limsup_{h to infty} int_K |nabla u_h(x)| dx leq P(E;K)$$
for all compact sets $K subset mathbb{R}^{n}$. Now fix a compact $J subset mathbb{R}$. I have proved that if $T in C^1_c(mathbb{R})$ satisfies $|T| leq 1$ and $J supset spt T$, then for a.e. $z in mathbb{R}^{n-1}$ it holds
$$ left| int_{E_z} T'(t) , dt right| leq liminf_{h to infty} int_{J} |nabla u_h (z,t)| , dt .$$
Taking the sup among the functions $T in C^1_c(mathbb{R})$ with $|T| leq 1$ and $J supset spt T$ and integrating on a compact set $H subset mathbb{R}^{n-1}$, we get
$$ int_H sup left{ left| int_{E_z} T' right| : T in C^1_c(mathbb{R}), , |T| leq 1 , , J supset spt T right} leq liminf_{h to infty} int_{H times J} |nabla u_h| leq P(E; H times J ) leq P(E) < infty. $$
If the integral of a function is finite then the function is a.e. finite. Hence for a.e. $z$ we have
$$ M_{J} := sup left{ left| int_{E_z} T' right| : T in C^1_c(mathbb{R}), , |T| leq 1 , , J supset spt T right} < infty.$$
This proves that the set has locally finite perimeter but I can't find a uniform bound for $M_{J}$.
Any help would be really appreciated!
real-analysis integration measure-theory geometric-measure-theory
asked 2 days ago
Hermione
235
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up vote
3
down vote
favorite
Before explaining my problem, I recall the definitions:
Let $E subset mathbb{R}^n$ be a Lebesgue measurable set. We say that $E$ is a set of locally finite perimeter if for every compact set $K subset mathbb{R}^n$ it holds that
begin{equation*} label{eq:deflocfiniteperimeter}
M_K := sup left{ int_{E} div, T(x) ,dx : T in C^1_c( mathbb{R}^n; mathbb{R}^n), spt ,T subset K, | T | leq 1 right} < infty.
end{equation*}
Moreover, if $$
supleft{M_K:K subset mathbb{R}^n, Ktext{compact}right}< infty,$$
then we say that $E$ is a set of finite perimeter.
Now suppose that $E$ is a set of finite perimeter. I have to prove that for $mathcal{L}^{n-1}$-a.e. $z in mathbb{R}^{n-1}$ the vertical slice $E_z subset mathbb{R}$ is a set of finite perimeter; where, by definition, $E_z := { t in mathbb{R} : (z,t ) in E}$.
I can only prove that for a.e. $z in mathbb{R}^{n-1}$ the vertical slice is a set of LOCALLY finite perimeter. Here I descrive my attempt:
Let $rho$ is a regulatizing kernel and consider the sequence $ u_{h} : = chi_E * rho_{1/h}$: I already know (from a previous result) that
$$ limsup_{h to infty} int_K |nabla u_h(x)| dx leq P(E;K)$$
for all compact sets $K subset mathbb{R}^{n}$. Now fix a compact $J subset mathbb{R}$. I have proved that if $T in C^1_c(mathbb{R})$ satisfies $|T| leq 1$ and $J supset spt T$, then for a.e. $z in mathbb{R}^{n-1}$ it holds
$$ left| int_{E_z} T'(t) , dt right| leq liminf_{h to infty} int_{J} |nabla u_h (z,t)| , dt .$$
Taking the sup among the functions $T in C^1_c(mathbb{R})$ with $|T| leq 1$ and $J supset spt T$ and integrating on a compact set $H subset mathbb{R}^{n-1}$, we get
$$ int_H sup left{ left| int_{E_z} T' right| : T in C^1_c(mathbb{R}), , |T| leq 1 , , J supset spt T right} leq liminf_{h to infty} int_{H times J} |nabla u_h| leq P(E; H times J ) leq P(E) < infty. $$
If the integral of a function is finite then the function is a.e. finite. Hence for a.e. $z$ we have
$$ M_{J} := sup left{ left| int_{E_z} T' right| : T in C^1_c(mathbb{R}), , |T| leq 1 , , J supset spt T right} < infty.$$
This proves that the set has locally finite perimeter but I can't find a uniform bound for $M_{J}$.
Any help would be really appreciated!
real-analysis integration measure-theory geometric-measure-theory
asked 2 days ago
Hermione
235
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Before explaining my problem, I recall the definitions:
Let $E subset mathbb{R}^n$ be a Lebesgue measurable set. We say that $E$ is a set of locally finite perimeter if for every compact set $K subset mathbb{R}^n$ it holds that
begin{equation*} label{eq:deflocfiniteperimeter}
M_K := sup left{ int_{E} div, T(x) ,dx : T in C^1_c( mathbb{R}^n; mathbb{R}^n), spt ,T subset K, | T | leq 1 right} < infty.
end{equation*}
Moreover, if $$
supleft{M_K:K subset mathbb{R}^n, Ktext{compact}right}< infty,$$
then we say that $E$ is a set of finite perimeter.
Now suppose that $E$ is a set of finite perimeter. I have to prove that for $mathcal{L}^{n-1}$-a.e. $z in mathbb{R}^{n-1}$ the vertical slice $E_z subset mathbb{R}$ is a set of finite perimeter; where, by definition, $E_z := { t in mathbb{R} : (z,t ) in E}$.
I can only prove that for a.e. $z in mathbb{R}^{n-1}$ the vertical slice is a set of LOCALLY finite perimeter. Here I descrive my attempt:
Let $rho$ is a regulatizing kernel and consider the sequence $ u_{h} : = chi_E * rho_{1/h}$: I already know (from a previous result) that
$$ limsup_{h to infty} int_K |nabla u_h(x)| dx leq P(E;K)$$
for all compact sets $K subset mathbb{R}^{n}$. Now fix a compact $J subset mathbb{R}$. I have proved that if $T in C^1_c(mathbb{R})$ satisfies $|T| leq 1$ and $J supset spt T$, then for a.e. $z in mathbb{R}^{n-1}$ it holds
$$ left| int_{E_z} T'(t) , dt right| leq liminf_{h to infty} int_{J} |nabla u_h (z,t)| , dt .$$
Taking the sup among the functions $T in C^1_c(mathbb{R})$ with $|T| leq 1$ and $J supset spt T$ and integrating on a compact set $H subset mathbb{R}^{n-1}$, we get
$$ int_H sup left{ left| int_{E_z} T' right| : T in C^1_c(mathbb{R}), , |T| leq 1 , , J supset spt T right} leq liminf_{h to infty} int_{H times J} |nabla u_h| leq P(E; H times J ) leq P(E) < infty. $$
If the integral of a function is finite then the function is a.e. finite. Hence for a.e. $z$ we have
$$ M_{J} := sup left{ left| int_{E_z} T' right| : T in C^1_c(mathbb{R}), , |T| leq 1 , , J supset spt T right} < infty.$$
This proves that the set has locally finite perimeter but I can't find a uniform bound for $M_{J}$.
Any help would be really appreciated!
real-analysis integration measure-theory geometric-measure-theory
asked 2 days ago
Hermione
235
Before explaining my problem, I recall the definitions:
Let $E subset mathbb{R}^n$ be a Lebesgue measurable set. We say that $E$ is a set of locally finite perimeter if for every compact set $K subset mathbb{R}^n$ it holds that
begin{equation*} label{eq:deflocfiniteperimeter}
M_K := sup left{ int_{E} div, T(x) ,dx : T in C^1_c( mathbb{R}^n; mathbb{R}^n), spt ,T subset K, | T | leq 1 right} < infty.
end{equation*}
Moreover, if $$
supleft{M_K:K subset mathbb{R}^n, Ktext{compact}right}< infty,$$
then we say that $E$ is a set of finite perimeter.
Now suppose that $E$ is a set of finite perimeter. I have to prove that for $mathcal{L}^{n-1}$-a.e. $z in mathbb{R}^{n-1}$ the vertical slice $E_z subset mathbb{R}$ is a set of finite perimeter; where, by definition, $E_z := { t in mathbb{R} : (z,t ) in E}$.
I can only prove that for a.e. $z in mathbb{R}^{n-1}$ the vertical slice is a set of LOCALLY finite perimeter. Here I descrive my attempt:
Let $rho$ is a regulatizing kernel and consider the sequence $ u_{h} : = chi_E * rho_{1/h}$: I already know (from a previous result) that
$$ limsup_{h to infty} int_K |nabla u_h(x)| dx leq P(E;K)$$
for all compact sets $K subset mathbb{R}^{n}$. Now fix a compact $J subset mathbb{R}$. I have proved that if $T in C^1_c(mathbb{R})$ satisfies $|T| leq 1$ and $J supset spt T$, then for a.e. $z in mathbb{R}^{n-1}$ it holds
$$ left| int_{E_z} T'(t) , dt right| leq liminf_{h to infty} int_{J} |nabla u_h (z,t)| , dt .$$
Taking the sup among the functions $T in C^1_c(mathbb{R})$ with $|T| leq 1$ and $J supset spt T$ and integrating on a compact set $H subset mathbb{R}^{n-1}$, we get
$$ int_H sup left{ left| int_{E_z} T' right| : T in C^1_c(mathbb{R}), , |T| leq 1 , , J supset spt T right} leq liminf_{h to infty} int_{H times J} |nabla u_h| leq P(E; H times J ) leq P(E) < infty. $$
If the integral of a function is finite then the function is a.e. finite. Hence for a.e. $z$ we have
$$ M_{J} := sup left{ left| int_{E_z} T' right| : T in C^1_c(mathbb{R}), , |T| leq 1 , , J supset spt T right} < infty.$$
This proves that the set has locally finite perimeter but I can't find a uniform bound for $M_{J}$.
Any help would be really appreciated!
real-analysis integration measure-theory geometric-measure-theory
real-analysis integration measure-theory geometric-measure-theory
asked 2 days ago
Hermione
235
asked 2 days ago
Hermione
235
edited 2 days ago
asked 2 days ago
Hermione
235
asked 2 days ago
Hermione
235
asked 2 days ago
Hermione
235
235
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