I'm trying to prove the following;
$W^{k,p}_{0} (\Bbb{R^2})$ is scale invariant for the the functional $E[u] = \int_{\Bbb{R}^2} |\delta_tu(0, x)| + |\delta_{xx}u(0, x)| dx $;
I have got the result
$E[u] = E[u_\lambda]$ where $u_\lambda(t,x) = u(\lambda^{-2}t, \lambda^{-1}x)$ for $\lambda \in (0, \infty)$
I also know that $W^{k,p}_{0}$ functions can be interpreted as the $W^{k,p}$ functions whose derivatives upto the $(k-1)^{th}$ order vanish on the boundary. In this case, $k=2, p=1$ so I must proof that $W^{2,1}_{0}$ is scale invariant. The part I don't get is where exactly the derivative of order 1 vanishes in this case? Here $u$ is a $\Bbb{C}^4(\Bbb{R}\times \Bbb{R^2})$ solution to the thin plate equation.
Also, how does only lower order derivatives in $W^{k,p}_{0}$ can vanish on boundary but the higher order derivative does not have this boundary vanishing property?