# 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?

(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).

• Thanks but what does it mean physically. Is there an interpretation. What is the motivating behind this?
– Jama
Dec 26, 2020 at 21:52
• @Jama physically I would say the ellipticity condition produces operators which are "close enough" to the laplacian, and therefore might model similar physical phenomena such as the diffusion. Jan 4, 2021 at 8:29