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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.

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I think the following lecture notes could be useful for your first two points (not the third, which I'm not familiar with unfortunately): It's only for capacity on $\mathbb{C}$ though. Also, for intuition: the capacity of a set is defined in a way to mimic the concept of capacity of a capacitator in physics/electrical engineering: if a set has positive capacity, the condensator obtained by having a perfect conductor of that set has positive capacity. This should help for calculating examples. A condensator also has the mentioned property of the boundary. – Jan Jitse Venselaar Nov 1 '12 at 18:26
up vote 4 down vote accepted

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.

  1. 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.

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