For some constant $\theta>0$ we have $$\frac{\theta}{2}\int_U |Du|^2 dx\le B[u,u]+C\int_U u^2dx$$ for some appropriate constant $C$. In addition we recall from Poincare's inequality that $$||u||_2\le C'||Du||_2$$ It easily follows that $$\beta ||u||_{H_0^1(U)}^2\le B[u,u]+\gamma||u||_{L^2(U)}^2$$ for appropriate constants $\beta>0,\gamma\ge 0$.
$$B[u,v]:=\int_U \sum_{i,j=1}^n a^{ij}u_{x_i}v_{x_j}+\sum_{i=1}^n b^i u_{x_i}v+cuv\, dx$$
This is the final part of the proof for the energy estimate in Evans PDE text, page 318-319. However, I don't see how to get the final inequality using Poincare's inequality. I would greatly appreciate any help.