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The meta-question is: understanding by how much a set fails to have full capacity. I will pin it down to some concrete, although non-exhaustive, questions in a reasonably simple framework. Let ${\mathbb D}_r$ be the disc in the plane centered at the origin and having radius $r\in(0,1)$ and ${\mathbb S}=\partial{\mathbb D}_1$. Consider subsets $E=\cup_i I_i\subseteq{\mathbb S}$ which are finite unions of disjoint closed arcs and define the following quantities: $$ \text{Cap}_1(E)=\inf\left\{\int_{{\mathbb D}_1}|\nabla u|^2dxdy:\ u|_{{\mathbb D}_{1/2}}=0,\ u|_E\ge1\right\}; $$ $$ \text{Cap}_2(E)=\sup\left\{\frac{\mu(E)^2}{\int_0^{2\pi}\left(\int_0^{2\pi}\frac{d\mu(s)}{|t-s|^{1/2}}\right)^2d\mu(t)}:\ \text{supp}(\mu)\subseteq E\right\}; $$ $$ \text{Cap}_3(E)=\sup\left\{\frac{\mu(E)^2}{\int_0^{2\pi}\int_0^{2\pi}\log\frac{2}{|e^{is}-e^{it}|}d\mu(s)d\mu(t)}:\ \text{supp}(\mu)\subseteq E\right\}. $$ These are three avatars of logarithmic capacity, this meaning that $$ \text{Cap}_1(E)\approx\text{Cap}_2(E)\approx\text{Cap}_3(E). $$ (Here $\approx$ means that the term on the left is bounded from below and above by a positive multiple of the term on the right).

There are many other quantities that are equivalent to $\text{Cap}_j(E)$ and which involve Green's functions, holomorphic functions, stochastic processes, dyadic objects...

The question is: which equivalences hold for the capacity deficit? For instance, is it true that: $$ \text{Cap}_1({\mathbb S})-\text{Cap}_1(E)\approx\text{Cap}_2({\mathbb S})-\text{Cap}_2(E), $$ or that $$ \text{Cap}_2({\mathbb S})-\text{Cap}_2(E)\approx\text{Cap}_3({\mathbb S})-\text{Cap}_3(E)? $$ I have been thinking on and off on this for a few years. The motivation is that, in the dyadic case, some estimates for the capacity of a condenser depend on this capacity deficit and a better understanding of it might lead to the extension of the estimates to classical Potential Theory. Here is a reference: https://arxiv.org/abs/1108.5325. For the equivalence of dyadic and classical capacity see: Benjamini, Itai; Peres, Yuval, Random walks on a tree and capacity in the interval. Ann. Inst. H. Poincaré Probab. Statist. 28 (1992), no. 4, 557–592.

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    $\begingroup$ In first question do you really want to mix Cap_1 and Cap_2 on LHS? and on RHS ? $\endgroup$ Jun 10, 2019 at 13:08
  • $\begingroup$ My typo: I edit right away. Thanks. $\endgroup$ Jun 10, 2019 at 13:44
  • $\begingroup$ My very uneducated guess would be that the deficit is simply comparable to the Lebesgue measure of $\mathbb{S} \setminus E$. Would that agree with your intuition? $\endgroup$ Jun 11, 2019 at 9:25
  • $\begingroup$ Well, not really: even using finite unions of arcs, you can construct subsets of the circle having arbitrarily small Lebesgue measure and capacity as close as you wish to that of the unit circle. You distribute the (small) length among many arcs of the same length and uniformly distributed in the circle. $\endgroup$ Jun 11, 2019 at 9:57
  • $\begingroup$ @NicolaArcozzi: Ah, right! Of course. $\endgroup$ Jun 11, 2019 at 10:30

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