I'm currently reading "Partial differential equation" by L.C. Evans (1st ed). On page 382, formula (25), Evans claimed that $||u_m||_{H_{0}^{1}}^{2}\leq A [u_m,u_m;t]$ by applying the uniform hyperbolicity inequality, i.e., $\int|Du|^{2}dx\leq A[u,u;t]$ where $A [u,v;t]=\int\sum_{{i}{j}}a^{ij}u_{x_i}v_{x_j} dx$.
I don't understand how he drived the above inequality, since $||u_m||_{H_{0}^{1}}^{2}$ involves also $L^2$ norm of $u$, and I can't see why the $L^2$ norm is also bounded by $A[u,u;t]$.
Any explainations?