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

Let $G$ be a domain assumed smooth enough. I want to show that the mean value $m$ is minimizing $ m \rightarrow \| f-m\|_{ L^2(G)} $ for $ f \in L^2(G)$. Is it unique? Is it allowed to derive under the integral?

share|cite|improve this question

There is no need to derivate inside the integra. To minimize $\|f-m\|_2$ is equivalent to minimize $$ \int_G(f(x)-m)^2\,dx=\int_Gf(x)^2\,dx-2\,m\int_Gf(x)\,dx+m^2\int_Gdx. $$

share|cite|improve this answer

Your Answer


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

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