Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I already asked this on M.SE, but get no answers.

Is there a (simple) example of a finely open set (i.e. w.r.t. the fine topology in potential theory) $O$ in $\mathbb R^n$, $n \ge 2$, which is not open up to a polar set (i.e. zero capacity), i.e., there does not exist a polar set $M$, such that the symmetric difference of $O$ and $M$ is open?

I found some examples of finely open sets (e.g. constructed using the Lebesgue spine), but all were "almost" open (in the above sense).

I use the following definition of the ($H_0^1$)-capacity: \begin{equation*} \operatorname{cap}(A) = \inf\big( \|\nabla v\|_{L^2(\Omega)}^2 : v \in H_0^1(\Omega) \text{ and } v \ge 1 \text{ on a neighbourhood of } A\big). \end{equation*} Here, $\Omega$ may be $\mathbb R^n$ or a bounded, open set.

share|improve this question

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.