convergence of a fraction containing Lebesgue measures and perimeters
Let $epsilon>0$ and consider the ellipsoid $$E_epsilon:={(x_1,dots x_n)inmathbb{R}^n mid frac{x_1^2}{1+epsilon}+x_2^2(1+epsilon)+x_3^2+dots +x_n^2le 1}.$$
Denoting with $P(E)$ the perimeter of a Borel set $Esubsetmathbb{R}^n$ (see https://en.wikipedia.org/wiki/Caccioppoli_set) and with $|E|$ the n-dimensional Lebesgue measure of $E$,
Why does $$limlimits_{epsilonto 0}frac{(P(E_epsilon)-P(B_1(0))|B_1(0)|^2}{P(B_1(0))|E_epsilon triangle B_1(0)|^2}$$ exist?
$E_epsilon triangle B_1(0)$ denotes the symmetric difference of $E_epsilon$ with the closed unit ball $B_1(0)$ in $mathbb{R}^n$. I know that $|E_epsilon triangle B_1(0)|to 0$ for $epsilonto 0$ and I know that the perimeter is lower semicontinuous w.r.t. the $L^1$-convergence of the characteristic functions of the sets. However, I don't know how to prove that the limit exists.
limits measure-theory lebesgue-measure geometric-measure-theory
asked 2 days ago
TheAppliedTheoreticalOneTheAppliedTheoreticalOne
645
add a comment |
Let $epsilon>0$ and consider the ellipsoid $$E_epsilon:={(x_1,dots x_n)inmathbb{R}^n mid frac{x_1^2}{1+epsilon}+x_2^2(1+epsilon)+x_3^2+dots +x_n^2le 1}.$$
Denoting with $P(E)$ the perimeter of a Borel set $Esubsetmathbb{R}^n$ (see https://en.wikipedia.org/wiki/Caccioppoli_set) and with $|E|$ the n-dimensional Lebesgue measure of $E$,
Why does $$limlimits_{epsilonto 0}frac{(P(E_epsilon)-P(B_1(0))|B_1(0)|^2}{P(B_1(0))|E_epsilon triangle B_1(0)|^2}$$ exist?
$E_epsilon triangle B_1(0)$ denotes the symmetric difference of $E_epsilon$ with the closed unit ball $B_1(0)$ in $mathbb{R}^n$. I know that $|E_epsilon triangle B_1(0)|to 0$ for $epsilonto 0$ and I know that the perimeter is lower semicontinuous w.r.t. the $L^1$-convergence of the characteristic functions of the sets. However, I don't know how to prove that the limit exists.
limits measure-theory lebesgue-measure geometric-measure-theory
asked 2 days ago
TheAppliedTheoreticalOneTheAppliedTheoreticalOne
645
add a comment |
Let $epsilon>0$ and consider the ellipsoid $$E_epsilon:={(x_1,dots x_n)inmathbb{R}^n mid frac{x_1^2}{1+epsilon}+x_2^2(1+epsilon)+x_3^2+dots +x_n^2le 1}.$$
Denoting with $P(E)$ the perimeter of a Borel set $Esubsetmathbb{R}^n$ (see https://en.wikipedia.org/wiki/Caccioppoli_set) and with $|E|$ the n-dimensional Lebesgue measure of $E$,
Why does $$limlimits_{epsilonto 0}frac{(P(E_epsilon)-P(B_1(0))|B_1(0)|^2}{P(B_1(0))|E_epsilon triangle B_1(0)|^2}$$ exist?
$E_epsilon triangle B_1(0)$ denotes the symmetric difference of $E_epsilon$ with the closed unit ball $B_1(0)$ in $mathbb{R}^n$. I know that $|E_epsilon triangle B_1(0)|to 0$ for $epsilonto 0$ and I know that the perimeter is lower semicontinuous w.r.t. the $L^1$-convergence of the characteristic functions of the sets. However, I don't know how to prove that the limit exists.
limits measure-theory lebesgue-measure geometric-measure-theory
asked 2 days ago
TheAppliedTheoreticalOneTheAppliedTheoreticalOne
645
Let $epsilon>0$ and consider the ellipsoid $$E_epsilon:={(x_1,dots x_n)inmathbb{R}^n mid frac{x_1^2}{1+epsilon}+x_2^2(1+epsilon)+x_3^2+dots +x_n^2le 1}.$$
Denoting with $P(E)$ the perimeter of a Borel set $Esubsetmathbb{R}^n$ (see https://en.wikipedia.org/wiki/Caccioppoli_set) and with $|E|$ the n-dimensional Lebesgue measure of $E$,
Why does $$limlimits_{epsilonto 0}frac{(P(E_epsilon)-P(B_1(0))|B_1(0)|^2}{P(B_1(0))|E_epsilon triangle B_1(0)|^2}$$ exist?
$E_epsilon triangle B_1(0)$ denotes the symmetric difference of $E_epsilon$ with the closed unit ball $B_1(0)$ in $mathbb{R}^n$. I know that $|E_epsilon triangle B_1(0)|to 0$ for $epsilonto 0$ and I know that the perimeter is lower semicontinuous w.r.t. the $L^1$-convergence of the characteristic functions of the sets. However, I don't know how to prove that the limit exists.
limits measure-theory lebesgue-measure geometric-measure-theory
limits measure-theory lebesgue-measure geometric-measure-theory
asked 2 days ago
TheAppliedTheoreticalOneTheAppliedTheoreticalOne
645
asked 2 days ago
TheAppliedTheoreticalOneTheAppliedTheoreticalOne
645
edited 2 days ago
TheAppliedTheoreticalOne
asked 2 days ago
TheAppliedTheoreticalOneTheAppliedTheoreticalOne
645
asked 2 days ago
TheAppliedTheoreticalOneTheAppliedTheoreticalOne
645
asked 2 days ago
TheAppliedTheoreticalOneTheAppliedTheoreticalOne
645
645
add a comment |
add a comment |
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