Poincaré-inequality

Question

I learned this week about sobolev-spaces and thereby my prof. claimed the truth of an inequality (without proof) that seemed to him very clear and easy to see but to me it was not clear at all. He used for $u \in H^1(0,1)$ that there is a constant C > 0, such that

$\|u\|_{L_2((0,1))} \leq C \cdot (\bar{u}^2 + |u|_{H^1(0,1)}^2)^{1/2}$

holds.

Notation: $\bar{u} := \int_{0}^{1}u(x)~dx$,

$|u|_{H^1(0,1)}$ is the seminorm on $H^1$, i.e. $|u|_{H^1(0,1)} = (\int_{0}^{1}(u'(x))^2~dx)^{1/2}$

I used the search in this forum and found out that in most cases similar statements have been proven by use of contradiction and a so called "Rellich–Kondrachov theorem" I've never heard before. I would like to know if there is an easy way to perhaps prove this in a direct way and if yes could someone give me a hint how to start? Thanks a lot!!!

Answer 1

Here is a sketch for a proof. Depending on what you already know, you might skip the first step.

1. Show that $u$ is absolutely continuous with $u(x) = \int_0^x u'(s) \,\mathrm{d}s$.
2. There is $\hat x$ with $u(\hat x) = \bar u$.
3. Use $u(x) = u(\hat x) + \int_{\hat x}^x u'(s) \, \mathrm{d}s$ to get an estimate for $\|u\|_{L^2(0,1)}$.