I was reading Evans' PDE,in the corresponding chapters Evans use elliptic energy estimate and Lax-Milgram theorem to prove the existence of uniformly elliptic equation and parabolic and hyperbolic equations.For the elliptic operator $Lu=-(a^{ij}u_{x_i})_{x_j}+b_iu_{x_i}+cu,$where $a_{ij}$ satisfies the uniform condition: $\sum a_{ij}\xi_i\xi_j\geq\theta\Vert\xi\Vert^2,$for $\theta\geq0$.Then we have the energy estimate:
$\beta||u||_{H^1_0(U)}^2\leq B[u,u]+\gamma||u||_{L^2(U)}^2,$for all $u,v\in H_0^1(U),\beta>0,\gamma\geq0$.
Here the bilinear form $B[\ ,\ ]$ is associated to the elliptic oprator $L$. This is enough to prove the existence of elliptic equation,no problem there.
But for the parabolic and hyperbolic equation,we still have the uniform parabolic(hyperbolic) condition.But we need a slightly different energy estimate:
$\beta||u||_{H^2(U)}^2\leq(Lu,-\Delta u)+\gamma||u||_{L^2(U)}^2,U\in H^2(U)\cap H_0^1(U),\beta>0,\gamma\geq0.$
Here we may pose some more conditions for this estimate in the very situation,$u$ is smooth,$\Delta u=0 $ on $\partial U$.To prove this,we may integrate-by-part twice and use the trace inequality :$||Tu||_{L^p(\partial U)} \leq ||u||_{W^{1,p}(U)}$ the main obstacle lies in this very inequality as we get a $L_p$ norm of third-order derivative of $u$ which cannot be removed and I don't know how to overcome this.
