While I was studying the book *Variation et Optimisation de formes* by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for every compact by:

$$ \operatorname{cap}(K)=\inf \lbrace \|u\|_{H^1(\Bbb{R}^N)} : v\in C_0^\infty(\Bbb{R}^N), v \geq 1 \text{ on }K\rbrace $$

The definition can be extended to open sets and then to every set of $\Bbb{R}^N$, relative capacity with respect to an open set can be defined by restricting the integral and the smooth function space to an open set D, etc.

The capacity has some strange properties which are unnatural at a first sight, like the fact that the capacity of $\partial K$ is the same as the capacity of $K$ for a compact $K$.

I want to understand better what capacity really means, and for that I tried to find all sort of books about potential theory (even the ones referred in the mentioned book), and all seem to have the same way of dealing with the subject: the setting is very general and abstract and the definition presented above just as a particular case.

Do you know any book, article or course notes which deal with this specific capacity in detail explaining:

the definition and the intuition behind the capacity;

examples of capacity computation for simple sets (using capacitary potentials);

the connection between the capacity and the Sobolev spaces ?

In the mentioned book the study of capacity is made in section 3.3. It contains all the definitions and all the needed properties of the capacity, but I still feel that I need a better understanding. That's why I asked this question.