Understanding Elliptic Operators

I don't have a strong background in partial differential equations so some of these questions might be quite basic. I first want to give some definitions which I am using.

Definition:

We define a second-order elliptic partial differential equation:

$Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+F=0$ where $B^{2}-AC<0$. Assuming $u_{xy} = u_{yx}$.

We define a uniformly elliptic operator in the following way:

Definition:

A partial differential operator $L$ is (uniformly) elliptic if there exists a constant $\theta>0$ such that $\sum_{i,j=1}^{\infty}a^{i,j}(x)\xi_{i}\xi_{j} \geq \theta|\xi|^{2}$

for a.e. $x \in U$ and all $\xi \in \mathbb{R}^{n}$

1.What is the connection between elliptic operators and elliptic partial differential equations? 2.What is the importance of this property $\sum_{i,j=1}^{\infty}a^{i,j}(x)\xi_{i}\xi_{j} \geq \theta|\xi|^{2}$ which is used in the definition of uniformly elliptic operators above? 3.Do uniformly elliptic operators always give you a linear partial differential equation?

Thanks for any assistance.

1. The constant matrix $a = (a^{i,j}) = \left[\begin{matrix} A & B \\ B & C \end{matrix}\right]$ satisfies the ellipticity condition if and only if $A>0$ and $AC-B^2>0$. For $n\times n$ matrices, the ellipticity condition is saying that $a$ is positive definite uniformly over $x$. And one criterion for positive definite is that the determinants of the $k\times k$ upper left hand corner matrices all have positive determinant. Note in the $2\times2$ case, the condition $A>0$ is not so important, because if $A<0$ you can simply multiply the whole equation by $-1$.

2. Ellipticity conditions are important because they guarantee uniqueness of solutions (although usually with other more benign hypotheses). This is a huge subject. I would recommend the book "Maximum Principles in Differential Equations" by Murray H. Protter, Hans F. Weinberger as a place to start.