I'm studying Elliptics Operators like this: $$Lu=a_{ij}(x)D_{ij}u+b_i(x)D_iu+c(x)u$$ for $u\in C^2(\Omega)\cap C(\overline{\Omega})$.
I want to know what the difference when:
- $L$ is elliptic in $\Omega$
- $L$ is strongly elliptic in $\Omega$
- $L$ is uniformly elliptic in $\Omega$
Can someone explain this?
In the beggining of Chapter 3 of Gilbard-Trudinge book, there is the definition and a example. To make thing easy to you (and for others readers), I will post it here.
I - $L$ is said $\bf{Elliptic}$ is the matrix $A(x)=(a_{ij}(x))$ is positive, i.e. if $\lambda (x),\Lambda (x)$ denote repsectively the minimum and maximum eigenvalues of $A(x)$ then $$\tag{1} 0<\lambda (x)|\xi|^2\leq a_{ij}(x)\xi_I\xi_j\leq\Lambda (x)|\xi|^2,\ \forall\ \xi=(\xi_1,...,\xi_n)\in\mathbb{R}^n\setminus\{0\}$$
II - $L$ is said to be $\bf{\mbox{Strictly Elliptic or Strongly Elliptic} }$ if $\lambda (x)$ defined above is bounded below by some positive constant $\lambda_0$.
III - $L$ is said $\bf{\mbox{Uniformly Elliptic}}$ if $\frac{\Lambda(x)}{\lambda(x)}$ is bounded.
Examples:
$\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$ satisfies I,II and III. Note that $A(x)=I$ where $I$ is the identity matrix.
The operator (in two variables $(x,y)$) $\frac{\partial^2}{\partial x^2}+x_1\frac{\partial ^2}{\partial y^2}$ satisfies I, but not III in he half plane: $x_1>0$. On the other hand, it does satisfies III in strips of the form $(\alpha,\beta)\times\mathbb{R}$ where $0<\alpha,\beta<\infty$.
As you can see from the previous examples, conditions I,II and III depends on the operator and on the domain of definition.
Remark: I would like to note that the terms may change depending on the authors, so when you are reading something related with this, pay attention to definitions.