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I'm currently studying a book on shape optimization: Variation et optimisation de formes: Une analyse géométrique By Antoine Henrot, Michel Pierre. The book introduces at some point capacity, and uses this to define quasi-open sets, quasi continuous functions, and these results apply very nice to approximation theorems in Sobolev spaces. As a remark, this is one of the few mathematical concepts I searched on Google, and found nothing like a wikipedia article or some sites which contain good knowledge about this subject in a beginner's terms.

The capacity is defined for compact sets first like this:

For $K\subset \Bbb{R}^N$ compact, we denote $cap(K)=\inf \{ \|v\|_{H^1(\Bbb{R}^N)}^2 : v \in C_0^\infty(\Bbb{R}^N),\ v \geq 1 \text{ on }K\}<\infty$. $C_0^\infty$ is the space of smooth functions with compact support.

From here, capacity extends to open sets, by taking the supremum on the capacities of compact sets contained in an open set.

The relative capacity is defined for a compact $K$ subset of a bounded open set $D\subset \Bbb{R}^N$ by $cap_D(K)=\inf\{ \int_D |\nabla v|^2 : v \in C_0^\infty(D),\ v \geq 1 \text{ on }K\}<\infty$.

I understood pretty well the Lebesgue measure by finding a way to visualize it (surely, this is not hard). I would like to know if there is something similar for capacity. How could I visualize it or understand it? How it relates to the physical reality? For example, what does the capacity of a ball or box in $\Bbb{R}^3,\Bbb{R}^n$ mean (for Lebesgue measure is the "volume")?

I would like to know if there are some books which treat the capacity subject in a manner like measure theory books do it, with many proved properties and some study problems to understand it better.

How can I understand capacity related to some physical aspect (shape, smoothness, finess...)? What are some good references for a beginner in the field?

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Have you looked at the books by Gustave Choquet? He's the inventor and writes extremely well, I think. I'm pretty sure his three-volume Cours d'analyse contains what you're looking for, and he also has the book Theory of capacities: Research on modern potential theory and Dirichlet problem. –  t.b. Jun 30 '11 at 22:01
    
I haven't looked into those books. I'll search for them. Thank you. –  Beni Bogosel Jun 30 '11 at 22:06
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A more modern reference containing the basics would be §30 of Kechris, Classical descriptive set theory. –  t.b. Jun 30 '11 at 22:15
    
This is very nice too :) Thank you. –  Beni Bogosel Jun 30 '11 at 22:31
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I can recommend the book "Variational Analysis in Sobolev and BV spaces" by Attouch et. al. –  Dirk Jul 1 '11 at 12:38
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up vote 2 down vote accepted

Since there is no answer posted, I will post some references I found meanwhile, apart of those presented in the comments:

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