Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

For $\nu\in (\epsilon,1)$ with $0<\epsilon<1$, let $P_\nu:H_0^1(\Omega)\rightarrow H_0^1(\Omega)$ with $\Omega\subset \mathbb{R}^N$ bounded Lipschitz domain, be the projection operator onto the set $K_\nu=\{v\in H_0^1(\Omega): |\nabla v|_{\mathbb{R}^N}\leq \nu \quad a.e.\}$. Is there an $L>0$ such that $$|P_\nu w-P_\rho w|_{H_0^1(\Omega)}\leq L |\nu-\rho|,$$ for all $w\in H_0^1(\Omega)$? Does the answer depend on $N$?

share|cite|improve this question
As it is now the set $K_\nu$ is not well defined. I guess what you meant is $$K_\nu=\{u \in H_0^1(\Omega):\ |\nabla u|_{\mathbb{R}^N}\le \nu \ \text{ a.e. } \}.$$ – Mercy King Sep 3 '12 at 0:26
$Mercy, yes, I edited 40 minutes before the comment; it is strange that you saw the old version, though! Thanks anyway! – Nonliapunov Sep 3 '12 at 0:30

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.