In Evan's Partial Differential Equations, he writes
Then, he continues to write:
But I do not understand how he gets
$I[w] \geq \delta ||Dw||^q_{L^q(U)} - \gamma$.
I tried to write it out and I got
$\int_U L dx \geq \int_U \alpha |Dw|^q-\beta dx = \int_U \alpha |Dw|^q dx - \beta|U|$, but why is it that
$\int_U \alpha |Dw|^q dx = \delta ||Dw||^q_{L^q(U)}$ for some $\delta > 0$
If I write $Dw = (w_1,w_2,...,w_n)$ isn't $\int_U \alpha |Dw|^q dx$ equal to $\alpha \int_U (|w_1|^2+|w_2|^2+... |w_n|^2)^{q/2} dx$ whereas $\delta ||Dw||^q_{L^q(U)}$ is equal to $\delta (\int_U (|w_1|^q+|w_2|^q+... |w_n|^q)dx)^{1/q}$?
How are these two integrals equal?
All norms in $\mathbb{R}^n$ are equivalent. There is a constant $C>1$ (depending on $n$ and $q$) such that $$ \frac{1}{C}\,\Bigl(\sum_{k=1}^n|x_n|^q\Bigr)^{1/q}\le\Bigl(\sum_{k=1}^n|x_n|^2\Bigr)^{1/2}\le C\,\Bigl(\sum_{k=1}^n|x_n|^q\Bigr)^{1/q} $$
Note that the last line of your argument is incorrect. $\|Dw\|^q_{L^q(U)}$ is NOT equal to $(\int_U (|w_1|^q+|w_2|^q+... |w_n|^q)\,dx)^{1/q}$, but to $\int_U (|w_1|^q+|w_2|^q+... |w_n|^q)\,dx$.