Let $U$ be a bounded region in n-dimensional space. Let $u(x,t)$, with $x\in U$ and $t>0$ be smooth and zero on $\partial U$. I have to prove the estimate $$ \int_U |\nabla u|^2 \leq C \int_U |\Delta u|^2, $$ where $C$ is some constant and must be related to the first eigenvalue $\lambda_1$ of the Laplacian.
According to Evans PDE $\lambda_1=\min_{u \in H_0^1(U),u\neq 0} \frac{\int_{\Omega} |\nabla u|^2 \;dx}{\int_{\Omega} |u|^2 \; dx}$. I thought about using integration by parts to get $$ \int_U u_{x_i}u_{x_i} \; dx = -\int_U u u_{x_i x_i} \; dx, $$ summing over $i$ to get $$ \int_U |\nabla u|^2 = -\int_U u \Delta u, $$ then using Holder's inequality $$ -\int_U u \Delta u \leq \left( \int_U |u|^2 \right)^{\frac{1}{2}} \left( \int_U |\Delta u|^2 \right)^{\frac{1}{2}}. $$ I don't see how to go on from here (if I'm in the right direction).
The rest is easy. You just need to apply the Friedrichs inequality $$ \int_U |u|^2 dx \leq C_h \int_U |\nabla u|^2 dx \quad\forall\, u\in H^1(U)\colon\; u|_{\partial U}=0, $$ valid for any Lipschitz domain $U\subset\mathbb{R}^n$ bounded in at least one direction, i.e., condained in some layer of width $h>0$ (see also http://en.wikipedia.org/wiki/Friedrichs%27_inequality).