Perimeter of the n-ball
0
$begingroup$
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)...
