Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

Consider the Bolza problem

$$ \inf\left\{F(u)=\int\limits_0^1 ((1-u'^2)^2+u^2)\, dx, u\in W^{1,4}(0,1), u(0)=0=u(1)\right\}. $$

Show that $\inf F(u)=0$, but that it does not exist an $u_0$ with $F(u_0)=0$.

Hello! To my opinion the fist step is to show that $F(u)\geq 0~\forall~u$. This is easy, I think, because the integrand is always $\geq 0$ anf therefore the integral, which means $F(u)$.

Then I have to find a sequence with $\lim\limits_{n\to\infty}F(u_n)=0$.

Can anybody help to find such a sequence? I did not have an idea yet...

How can I construct such a sequence? I don' t see that.

Thank you very much for helping!

share|cite|improve this question

The first part is OK. For the second part, if there is a $u \in W^{1,4}(0,1)$ with $u(0)=0=u(1)$ such that $F(u)=0$. Then $u^2=0=1-(u')^2$ a.e. in $(0,1)$, i.e. $u=0$ a.e. in $(0,1)$, but $u'\ne 0$ a.e. in $(0,1)$. This is not possible, hence there is no such a $u$.

share|cite|improve this answer
Thank you. But the first part is not finished yet! I have to find a sequence $(u_n)$ with $\lim\limits_{n\to\infty}F(u_n)=0$, haven't I? – math12 Jan 21 '13 at 16:28
@math12 Use a triangular wave: a sequence of small zigzags with slope $\pm 1$. – user53153 Jan 21 '13 at 16:52
Do you mean $u_n(x)=\begin{cases}x, & x\in [0,\frac{1}{n})\\-x+\frac{n}{2}, & x\in [\frac{1}{n},\frac{2}{n})\\x-\frac{n}{2}, & x\in [\frac{2}{n},\frac{3}{n})\\-x+\frac{n}{4}, & x\in [\frac{3}{n},\frac{4}{n})\\...\\-x+1, & x\in [\frac{n-1}{n},1]\end{cases}$? – math12 Jan 21 '13 at 17:17
@math12 I think the terms you add should be something like $j/n$. Or you can define $u$ in one line: $$u(x)=\int_0^x \operatorname{sign}\sin 2\pi n t\,dt$$ – user53153 Jan 21 '13 at 20:39
Maybe you can explain me how you came to that? – math12 Jan 21 '13 at 20:44

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.