Perimeter of the n-ball
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My task is to calculate the perimeter of the 2-ball (n-ball) using De Giorgi theorem for BV functions.
By definition the perimeter of a Borel set $A$ is the $BV$-norm of its characteristic function $I_A$. De Giorgi theorem states that, if $f in L^1(mathbb{R}^n)$, then $Vert f Vert_{BV}=lim_{t rightarrow 0^+} int_{mathbb{R}^n}Vertnabla P_t f Vert dx$.
In the statement of the theorem $(P_t)_{t geq 0}$ is the Wiener semigroup, namely $P_t f(x)=int_{mathbb{R}^n} f(x+sqrt{t} y)frac{exp(-Vert y Vert^2/2)}{sqrt{2pi}}dy$.
Using change of variables in the integral it turns out to be useful to rewrite $P_t f(x)=int_{mathbb{R}^n} f(x-y)rho_t(y)dy=int_{mathbb{R}^n}f(y)rho_t(x-y)$, where $rho_t(y)=frac{e^{-frac{Vert yVert^2}{2t}}}{2 pi t}$.
In my case of interest $f=I_B$, where $B$ is (to start) the $2$-dimensional ball (later the $n$-dimensional one).
I tried to calculate $nabla P_t I_B$ exchanging derivative and integral, i.e. $frac{partial}{partial x_1}P_t I_B(x)=int_{B}frac{partial}{partial x_1}rho_t(x-y)dy$.
Now i have to integrate over the ball, so i'd like to use Gauss-Green. Using $F(x)=(rho_t(x-y),0)$, i should have $int_{B}frac{partial}{partial x_1}rho_t(x-y)dy=int_{partial B} F.dP$, where the last is the line integral of $F$ over the sphere of radius $r$. How could i go on from this point?
real-analysis integration probability-distributions
asked Dec 2 '18 at 17:24
AndroAndro
787
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My task is to calculate the perimeter of the 2-ball (n-ball) using De Giorgi theorem for BV functions.
By definition the perimeter of a Borel set $A$ is the $BV$-norm of its characteristic function $I_A$. De Giorgi theorem states that, if $f in L^1(mathbb{R}^n)$, then $Vert f Vert_{BV}=lim_{t rightarrow 0^+} int_{mathbb{R}^n}Vertnabla P_t f Vert dx$.
In the statement of the theorem $(P_t)_{t geq 0}$ is the Wiener semigroup, namely $P_t f(x)=int_{mathbb{R}^n} f(x+sqrt{t} y)frac{exp(-Vert y Vert^2/2)}{sqrt{2pi}}dy$.
Using change of variables in the integral it turns out to be useful to rewrite $P_t f(x)=int_{mathbb{R}^n} f(x-y)rho_t(y)dy=int_{mathbb{R}^n}f(y)rho_t(x-y)$, where $rho_t(y)=frac{e^{-frac{Vert yVert^2}{2t}}}{2 pi t}$.
In my case of interest $f=I_B$, where $B$ is (to start) the $2$-dimensional ball (later the $n$-dimensional one).
I tried to calculate $nabla P_t I_B$ exchanging derivative and integral, i.e. $frac{partial}{partial x_1}P_t I_B(x)=int_{B}frac{partial}{partial x_1}rho_t(x-y)dy$.
Now i have to integrate over the ball, so i'd like to use Gauss-Green. Using $F(x)=(rho_t(x-y),0)$, i should have $int_{B}frac{partial}{partial x_1}rho_t(x-y)dy=int_{partial B} F.dP$, where the last is the line integral of $F$ over the sphere of radius $r$. How could i go on from this point?
real-analysis integration probability-distributions
asked Dec 2 '18 at 17:24
AndroAndro
787
$endgroup$
add a comment |
$begingroup$
My task is to calculate the perimeter of the 2-ball (n-ball) using De Giorgi theorem for BV functions.
By definition the perimeter of a Borel set $A$ is the $BV$-norm of its characteristic function $I_A$. De Giorgi theorem states that, if $f in L^1(mathbb{R}^n)$, then $Vert f Vert_{BV}=lim_{t rightarrow 0^+} int_{mathbb{R}^n}Vertnabla P_t f Vert dx$.
In the statement of the theorem $(P_t)_{t geq 0}$ is the Wiener semigroup, namely $P_t f(x)=int_{mathbb{R}^n} f(x+sqrt{t} y)frac{exp(-Vert y Vert^2/2)}{sqrt{2pi}}dy$.
Using change of variables in the integral it turns out to be useful to rewrite $P_t f(x)=int_{mathbb{R}^n} f(x-y)rho_t(y)dy=int_{mathbb{R}^n}f(y)rho_t(x-y)$, where $rho_t(y)=frac{e^{-frac{Vert yVert^2}{2t}}}{2 pi t}$.
In my case of interest $f=I_B$, where $B$ is (to start) the $2$-dimensional ball (later the $n$-dimensional one).
I tried to calculate $nabla P_t I_B$ exchanging derivative and integral, i.e. $frac{partial}{partial x_1}P_t I_B(x)=int_{B}frac{partial}{partial x_1}rho_t(x-y)dy$.
Now i have to integrate over the ball, so i'd like to use Gauss-Green. Using $F(x)=(rho_t(x-y),0)$, i should have $int_{B}frac{partial}{partial x_1}rho_t(x-y)dy=int_{partial B} F.dP$, where the last is the line integral of $F$ over the sphere of radius $r$. How could i go on from this point?
real-analysis integration probability-distributions
asked Dec 2 '18 at 17:24
AndroAndro
787
$endgroup$
My task is to calculate the perimeter of the 2-ball (n-ball) using De Giorgi theorem for BV functions.
By definition the perimeter of a Borel set $A$ is the $BV$-norm of its characteristic function $I_A$. De Giorgi theorem states that, if $f in L^1(mathbb{R}^n)$, then $Vert f Vert_{BV}=lim_{t rightarrow 0^+} int_{mathbb{R}^n}Vertnabla P_t f Vert dx$.
In the statement of the theorem $(P_t)_{t geq 0}$ is the Wiener semigroup, namely $P_t f(x)=int_{mathbb{R}^n} f(x+sqrt{t} y)frac{exp(-Vert y Vert^2/2)}{sqrt{2pi}}dy$.
Using change of variables in the integral it turns out to be useful to rewrite $P_t f(x)=int_{mathbb{R}^n} f(x-y)rho_t(y)dy=int_{mathbb{R}^n}f(y)rho_t(x-y)$, where $rho_t(y)=frac{e^{-frac{Vert yVert^2}{2t}}}{2 pi t}$.
In my case of interest $f=I_B$, where $B$ is (to start) the $2$-dimensional ball (later the $n$-dimensional one).
I tried to calculate $nabla P_t I_B$ exchanging derivative and integral, i.e. $frac{partial}{partial x_1}P_t I_B(x)=int_{B}frac{partial}{partial x_1}rho_t(x-y)dy$.
Now i have to integrate over the ball, so i'd like to use Gauss-Green. Using $F(x)=(rho_t(x-y),0)$, i should have $int_{B}frac{partial}{partial x_1}rho_t(x-y)dy=int_{partial B} F.dP$, where the last is the line integral of $F$ over the sphere of radius $r$. How could i go on from this point?
real-analysis integration probability-distributions
real-analysis integration probability-distributions
asked Dec 2 '18 at 17:24
AndroAndro
787
asked Dec 2 '18 at 17:24
AndroAndro
787
asked Dec 2 '18 at 17:24
AndroAndro
787
asked Dec 2 '18 at 17:24
AndroAndro
787
asked Dec 2 '18 at 17:24
AndroAndro
787
787
add a comment |
add a comment |
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