Books about capacity theory 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.
 A: I think the best treatment of basic facts about capacity from the perspective of Sobolev spaces is in Chapter 4 of
L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions. Revised edition. Textbooks in Mathematics. CRC Press, Boca Raton, FL, 2015.
(MathSciNet review).
The book of Maz'ya (see Changyu Guo's answer) is very comprehensive, but difficult to read and I would not recommend it as an introduction. The book by Heinonen is about analysis on metric spaces so this is a different story and the book by Heinonen, Kilpelainen and Martio deals with a quite advanced nonlinear potential theory. This being said, if you want to learn basic results about capacity theory read Evans and Gariepy!
A: Maz'ya's book contains a fruitful treatment of Capacity and Weighted capacity and its relation with Sobolev spaces theory, in particular the (weighted) Sobolev inequality or Poincare inequality. Heinonen's book contains the treatment of modulus and capacity in metric setting. 


*

*Maz'ya, Vladimir Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 342. Springer, Heidelberg, 2011. xxviii+866 pp.


2.Heinonen, Juha Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001. x+140 pp.
3.Heinonen, Juha; Kilpeläinen, Tero; Martio, Olli Nonlinear potential theory of degenerate elliptic equations. Unabridged republication of the 1993 original. Dover Publications, Inc., Mineola, NY, 2006. xii+404 pp.
A: Also a very good book:
Title: Condenser Capacities and Symmetrization in Geometric Function Theory
Author(s): Vladimir N. Dubinin (auth.)
Publisher: Birkhäuser Basel
Year: 2014
ISBN: 978-3-0348-0842-2,978-3-0348-0843-9
