Poincare Inequality implies Equivalent Norms

Question

I am currently working through the subject of Sobolev Spaces using the book 'Partial Differential Equations' by Lawrence Evans. After the result proving the Poincare Inequality it says the following in the book(page 266.) "In view of the Poincare Inequality, on $W_{0}^{1,p}(U)$ the norm $||DU||_{L^{p}}$ is equivalent to $||u||_{W^{1,p}(U)}$, if $U$ is bounded." Do you know the argument behind this statement?

Answer 1

This applies to $1 \leq p < n$. You want constants $B_{1}$ and $B_{2}$ such that $B_{1}||u||_{W_{0}^{1,p}(U)} \leq ||Du||_{L^{p}(U)} \leq B_{2}||u||_{W_{0}^{1,p}}$. The inequality $||Du||_{L^{p}(U)} \leq B_{2}||u||_{W_{0}^{1,p}}$ is trivial.

From Gagliardo-Nirenberg-Sobolev Inequality we have: $||u||_{L^{p*}(U)} \leq C||Du||_{L^{p}(U)}$ where $p* := \frac{np}{n-p}$ $p* > p$.

We also have $||u||_{L^{q}(U)} \leq C||u||_{L^{p*}(U)}$ if $1\leq q \leq p*$ by Generalized Holder Inequality.

If we take $A = \frac{1}{c^{2}}$ and $q=p$ then we have $A||u||_{L^{p}} \leq ||Du||_{L^{p}}$ then since $||u||_{W_{0}^{1,p}} \leq A||u||_{L^{p}} + A||Du||_{L^{p}} \leq (1+A)||Du||_{L^{p}}$ it follows that $\frac{A}{A+1}||u||_{W_{0}^{1,p}} \leq ||Du||_{L^{p}}$

Take $B_{1} := \frac{A}{A+1}$.

The result follows from combining the two inequalities.

Answer 2

$$ \|Du\|_{L^p(U)} \le \|Du\|_{L^p(U)}+\|u\|_{L^p(U)} = \|u\|_{W_0^{1,p}(U)} \le (1+C)\,\|Du\|_{L^p(U)} $$ where $C$ is the constant in Poincare's inequality.