I am trying to understand the proof of a theorem of capacity theory

the theorem can be found in the book "nonlinear potential theory of degenerate elliptic equations" the authors are Juha Heinonen, Terio Kilpelainen and Olli MArtio

I am trying to underrstand the part (iv) of the theorem. the theorem is the theorem 2.2 in the page 28

2.2 theorem:

(part iv) If $K_i$ is a decreasing sequence of compact subsets of $\Omega \subset R^n$ with $K = \bigcap_{i} K_i$ ,then

$cap_{p , \theta} (K, \Omega) = lim_{i -> \infty} cap_{p , \theta} (K_i, \Omega)$

In the proof i am not understanding the part "When $i$ is large , the sets $K_i$ lie in the compact set $\{ u \geq 1- \epsilon \}$"

Any help will help me

Thanks !

(my english is terrible, sorry..)

  • $\begingroup$ Can you mention what the function $u$ is? $\endgroup$ – Umberto P. Mar 27 '13 at 2:42
  • $\begingroup$ Also you could try emailing Kilpelainen. Heinonen is deceased and as far as I know Martio is retired. $\endgroup$ – Umberto P. Mar 27 '13 at 3:30

I'll try to save Tero Kilpeläinen the trouble of fielding an email; besides, I don't think that authors are responsible for explaining basic analysis proofs found at the beginning of their PDE book.

The function $u$ is continuous on $\Omega$ and $u\ge 1$ on $K$. The sets $K_i\cap \{u\le 1-\epsilon\}$ form a nested sequence of compact sets. Since $K=\bigcap_i K_i$, it follows that $$\bigcap_i (K_i\cap \{u\le 1-\epsilon\}) = K\cap \{u\le 1-\epsilon\} = \varnothing$$ If the intersection of a nested sequence of compact sets is empty, then there exists $i_0$ that the sets are empty for $i\ge i_0$. Therefore, for $i\ge i_0$ we have $K_i\subset \{u> 1-\epsilon\}$.


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