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16h
comment Estimating the $(N-1)$- Hausdorff measure of $\Omega\cap \partial B(0,r)$ when $\lim_{r\to\infty} m(\Omega\cap B(0,r))/m(B(0,r))=0$.
What happens if I ask the following: Assuming the same condition, is it true that for almost every $\omega\in S(0,1)$, there exist $t_0>0$ such that, the set $r_{\omega,t_0}=\{t\omega, t\ge t_0\}$ satisfies $$\Omega\cap r_{\omega,t_0}=\emptyset?$$
16h
accepted Estimating the $(N-1)$- Hausdorff measure of $\Omega\cap \partial B(0,r)$ when $\lim_{r\to\infty} m(\Omega\cap B(0,r))/m(B(0,r))=0$.
16h
comment Estimating the $(N-1)$- Hausdorff measure of $\Omega\cap \partial B(0,r)$ when $\lim_{r\to\infty} m(\Omega\cap B(0,r))/m(B(0,r))=0$.
No, you are right. I just forget to multiply by $r$. Thank you again.
17h
comment Estimating the $(N-1)$- Hausdorff measure of $\Omega\cap \partial B(0,r)$ when $\lim_{r\to\infty} m(\Omega\cap B(0,r))/m(B(0,r))=0$.
Thank for you answer @Daniel. I cannot see why $(2)$ is true. For example, in the case $N=2$, it seem to me that $\mathcal{H}^{N-1}\left(\Omega \cap \partial B(0,r)\right)$ would be proportional to $\arctan{(1/r)}$. Maybe I am missgin something...
18h
revised Estimating the $(N-1)$- Hausdorff measure of $\Omega\cap \partial B(0,r)$ when $\lim_{r\to\infty} m(\Omega\cap B(0,r))/m(B(0,r))=0$.
added 7 characters in body
20h
asked Estimating the $(N-1)$- Hausdorff measure of $\Omega\cap \partial B(0,r)$ when $\lim_{r\to\infty} m(\Omega\cap B(0,r))/m(B(0,r))=0$.
1d
comment Relation between the Eigenvalue and the potentials.
What is $\Omega$? What is $u_i$? Is it in some Sobolev space? Is this a weak solution? Strong? What do you mean by $-\nabla\gamma_i\nabla u_i$?
1d
comment Shortest p-distance
For every $p$ other than $1$ or $\infty$, the ball is strictly convex.
Jul
21
asked Hopf lemma for generalized normal derivatives
Jul
21
reviewed Close proving closure of a subset
Jul
21
reviewed Close Alternate series
Jul
21
reviewed Close integrate $ \int \frac {x dx}{\sqrt {1+x^{10}} } $
Jul
21
comment Is this distributional laplacean a measure?
@Thisismuchhealthier. Let $\delta>0$ be small, $\Omega_\delta=\{x\in \Omega: d(x,\partial\Omega)<\delta\}$. Let $u_\delta(x)=u(x)$ for $x\in \Omega\setminus\Omega_\delta$ and $u_\delta(x)=u(\tau_\delta(x))d(x,\partial\Omega)/\delta$ for $x\in \Omega_\delta$, where $\tau_\delta(x)$ satisfies $d(x,\partial(\Omega\setminus\Omega_\delta)=d(x,\tau_\delta(x))$. Is true that $u_\delta \to u$ in $W^{1,2}(\Omega)$? If this is true, I think I can use $u_\delta$ to approximate $u$ by smooth, superharmonic functions.
Jul
21
comment Is this distributional laplacean a measure?
@Thisismuchhealthier. I see, and now I have the problem that $u_n$ is not superharmonic. So, in some way, we will have to consider the normal derivative of $u$ along the boundary.
Jul
21
comment Is this distributional laplacean a measure?
@Thisismuchhealthier. Thank you for your reply. All calculations are local, i.e., in $\Omega$, so I can't see where the generalized Laplacian along the boundary, would give some trouble. Could you point me out, where is the possible flaw?
Jul
21
revised Is this distributional laplacean a measure?
added 18 characters in body
Jul
20
reviewed Close Finding the value of this product
Jul
19
reviewed Close find this graph thing is very hard
Jul
19
reviewed Close i can not figure out probability.
Jul
19
reviewed Close real analysis the lime as x approaches epsilon