Let $U$ be the bounded smooth open subset of $\Bbb{R}^n$, with $u \in H^2 \cap H^1_0$. Let $L = \sum_{ij} (a_{ij}(x) u_{x^i})_{x^j} + \sum_k b_k(x) u_{x^k} + c(x) u$ be a general linear differential operator which is strictly elliptic with smooth coefficients. Prove:
$$\theta\|{-\Delta u}\|_{L^2(U)}^2 \leq (Lu,-\Delta u), $$
My attempt: For simplicity first assume $L = \sum a_{ij} u_{x^ix^j}$ in the non divergence form and further assume $u$ is sufficient smooth and everything vanish on the boundary,with $a_{ij}$ does not depend on $x$, we can expand the RHS as:
$$\int_U \sum_{i,j,k} a_{ij}u_{x^i x^j}u_{x^kx^k} dx = \sum_k \sum_{ij} \int_U a_{ij}u_{x^ix^k}u_{x^jx^k} dx\ge \theta\int_U |D^2u|^2 dx \ge \theta \|\Delta u\|_{L^2}^2$$
As desired , then I needs to consider more general case which seems very difficult to deduce.
More general also means there exist some boundary term when integrating by parts. And $u$ is not regular enough to integrate by parts.