ellipticity condition 
I believe (1.3) is in divergence form. My problem is that it is somewhat abstract for me and I'd like to see a few concrete examples.
I'd appreciate seeing an actual differential equation of this form with a functions $a_{ij}, b_i, c$ given values and with these values see the matrix formed by the $a_{ij}$
What does the ellipticity condition mean in words and why is it important?
PS: Is this the right place to ask clarification/understanding questions or is there a special place on SE Math for that?
 A: (1.3) such that (1.4) holds is the most general form of a (uniformly) elliptic operator. Notice that indeed the laplacian is elliptic since we can write it like in (1.3) with $a_{ij}=\delta_{ij}$, $b_i=0$ and $c=0$ and (1.4) holds trivially. One can construct other elliptic operators by modifying the laplacian operator such that (1.4) still holds and then adding any continuous functions as $b_i$ and $c$, this last part doesn't really play any role on the ellipticity condition.
The ellipticity condition might be very important in many contexts but specially I would like to point out that in order to prove the existence of weak solutions by means of the Lax-Milgram theorem is key. Let $\mathcal{L}u=f$ be our elliptic equation with
$$ \mathcal{L}u = -\sum_{i,j=1}^n \partial_i (a_{ij}(x)\partial_j u(x)) + \sum_{i=1}^n b_i(x)\partial_i u(x) + c(x)u(x). $$
If we write it in the weak formulation we get
$$ \underbrace{\sum_{i,j=1}^n \int_\Omega a_{ij}\partial_j u \partial_i\phi + \sum_{i=1}^n \int_\Omega b_i\partial_i u\phi + \int_\Omega cu\phi}_{B[u,\phi]} = \underbrace{\int_\Omega f\phi}_{\langle f,\phi \rangle} $$
for all $\phi\in C_c^\infty(\Omega)$. Then by Lax-Milgram we can prove existence of (weak) solutions if the bilinear form $B[\cdot,\cdot]$ is coercive, that is
$$ |B[u,u]| \geq C\|u\|^2. $$
Depending on the operator $\mathcal{L}$ ellipticity might be enough to prove it. In other cases not but still gives a good energy estimate that will be of help to prove well-posedness (see Evans' book Partial Differential Equations, Section 6.2, Theorem 2).
