Poincaré's Inequality on Sobolev Spaces in One Dimension

Question

The following is a version of Poincaré's inequality:
Let $I$ be a bounded interval, then there exists a constant $C$ dependent on $I$ such that $$\|u\|_{W^{1,p}(I)} \leq C\|u'\|_{L^p(I)} \ \ \ \ \forall u \in W_0^{1,p}(I).$$

According to the proof I have, this can be done by using the fact that
$$\|u\|_{{L^\infty}(I)} \leq \|u'\|_{L^1(I)}$$ and Hölder's inequality but I just cannot see it. Is anybody able to help me out?

Answer 1

Take $I$ to be $[a,b]$ and $x \in [a,b]$ then (because of the vanishing at the boundary assumption):

$$u(x)=\int_a^x u'(y) dy.$$

Thus

$$|u(x)| \leq \int_a^x |u'(y)| dy$$

so

$$\| u \|_{L^\infty} \leq \| u' \|_{L^1}$$

by taking suprema over $x$.

Next

$$\| u \|_{L^p} = \| u^p \cdot 1 \|_{L^1}^{1/p} \leq \| u^p \|^{1/p}_{L^\infty} \| 1 \|_{L^1}^{1/p}.$$

Note that it is here that you need that $I$ is bounded, so that $1 \in L^1$.

Finally, whatever definition of $\| u \|_{W^{1,p}}$ you take, you have

$$\| u \|_{W^{1,p}} \leq C \left ( \| u \|_{L^p} + \| u' \|_{L^p} \right ).$$

Put these pieces together.