0
$\begingroup$

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..)

$\endgroup$
  • $\begingroup$ 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
3
$\begingroup$

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 $K_i\subset \{u> 1-\epsilon\}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.