Energy estimates (Evans PDE Chapter 6): Sobolev norm is controlled by an elliptic operator

Question

I am looking at the energy estimates of 2nd-order elliptic PDE in Evans p.318. In the following, everything follows, except that I cannot understand how the final line obtained. When I apply poincare inequality, I ended up with $$\beta\|u\|_{H_0^1}^2\le B[u,u].$$

Please help.

Answer 1

One of forms of the Poincaré inequality says that for functions in $H^1_0$, the $L^2$ norm is bounded by a multiple of the $L^2$ norm of the gradient. Therefore, for $u\in H_0^1$, $$\|u\|_{H_0^1}^2 \le C_1\|Du\|_{L^2}^2 \tag{$*$}$$ At the end of page 318, the right hand side of $(*)$ is estimated by $B[u,u]+C\|u\|_{L^2}^2$. Thus, $$\|u\|_{H_0^1}^2 \le C_2B[u,u] +C_3\|u\|_{L^2}^2 $$