Prove the Laplacian operator is uniformly elliptic

Question

Given $$Lu=a_{ij}(x)D_{ij}u+b_i(x)D_iu+c(x)u$$ a partial differential operator $L$ is (uniformly) elliptic if there exists a constant $\theta>0$ such that $\sum_{i,j=1}^{n}a^{i,j}(x)\xi_{i}\xi_{j} \geq \theta|\xi|^{2}.$

I've read everywhere that the Laplacian operetor $\Delta$ is (uniformly) elliptic but i can't see why.
Can someone show me why?

Answer 1

The Laplacian is a constant coefficient operator with no off-diagonal terms (so $a^{ij}\equiv 0$, for $i\neq j$), so the result follows by considering $\theta=1$.