First of all, I am sorry if this is a too basic question, but I stumbled over this notion of ellipticity only very recently.

I am trying to understand the definition of ellipticity of systems due to Agmon-Douglis-Nirenberg, but I am lacking some intuition. For definiteness, let me work with an explicit example. Consider the Laplacian on $\mathbb{R}^{2}$, i.e. $\Delta:C^{\infty}(U)\to C^{\infty}(U)$ for some open $U\subset\mathbb{R}^{2}$ given by $\Delta u:=\partial_{x}^{2}u+\partial_{y}^{2}u$. Now, it is well known that this operator is elliptic: Its principal symbol is given by $$\sigma_{\mathrm{pr}}(x,\xi)=\xi_{1}^{2}+\xi_{2}^{2}$$ for all $(x,\xi)\in U\times\mathbb{R}^{2}$, which is invertible whenever $\xi\neq 0$. Now, lets rewrite this operator as 1st order system $$D\begin{pmatrix} u_{1}:=u \\\ u_{2}:=\partial_{x}u \\\ u_{3}:=\partial_{y}u\end{pmatrix}=\begin{pmatrix}0 &\partial_{x} &\partial_{y}\\\ \partial_{x} &-1 & 0\\\ \partial_{y} &0 &-1\end{pmatrix}\begin{pmatrix} u_{1} \\\ u_{2} \\\ u_{3}\end{pmatrix}$$

Now, the principal symbol of this operator is given by $$\sigma_{\mathrm{pr}}(x,\xi)=\begin{pmatrix}0 &\xi_{x} &\xi_{y}\\\ \xi_{x} &0 & 0\\\ \xi_{y} &0 &0\end{pmatrix}$$ which is not invertible, so $D$ is not elliptic in the usual sense. For this reason, Agmon-Douglis-Nirenberg introduced a refined definition of ellipticity, in which the fact that the entries of $D$ are of different degree is taken care of. So, one chooses integer weights $s_{i}$ for all the equations and $t_{j}$ for all the unknows such that $s_{i}+t_{j}\geq d_{ij}$ where $d_{ij}$ is the highest order of derivative of the jth unknown in the ith equation. Then, the weighted principal symbol of a differential operator $D=\sum_{\vert\alpha\vert\leq d}a_{\alpha}(x)\partial^{\alpha}$ is defined by

$${\sigma_{\mathrm{pr},w}(x,\xi)}_{ij}=\sum_{\vert\alpha\vert=s_{i}+t_{j}}(a_{\alpha}(x))_{ij}\xi^{\alpha}$$

An operator $D$ is then called "ADN-elliptic" if there exists a choice of weights such that the weighted principal symbol is invertible. Now, for the example above, suitable weights are $s_{1}=s_{2}=0,s_{3}=-1$ for equations and $t_{1}=2,t_{2}=t_{3}=1$ for unknows, so that the weighted symbol becomes $$\sigma_{\mathrm{pr},w}(x,\xi)=\begin{pmatrix}0 &\xi_{x} &\xi_{y}\\\ \xi_{x} &-1 & 0\\\ \xi_{y} &0 &-1\end{pmatrix}$$ which is invertible for $\xi\neq 0$.

What is the intuition for the choice of weights in general?

In the above example, the choice seems to be rather random. Is there some general strategy in choosing weights? In the end, the weights are there to deal with the fact that the different entries of the matrix operator $D$ are of different order. So is there a canonical choice related to the order of each entry? How does it work in the example above?

First of all, I am sorry if this is a too basic question, but I stumbled over this notion of ellipticity only very recently.

I am trying to understand the definition of ellipticity of systems due to Agmon-Douglis-Nirenberg, but I am lacking some intuition. For definiteness, let me work with an explicit example. Consider the Laplacian on $\mathbb{R}^{2}$, i.e. $\Delta:C^{\infty}(U)\to C^{\infty}(U)$ for some open $U\subset\mathbb{R}^{2}$ given by $\Delta u:=\partial_{x}^{2}u+\partial_{y}^{2}u$. Now, it is well known that this operator is elliptic: Its principal symbol is given by $$\sigma_{\mathrm{pr}}(x,\xi)=\xi_{1}^{2}+\xi_{2}^{2}$$ for all $(x,\xi)\in U\times\mathbb{R}^{2}$, which is invertible whenever $\xi\neq 0$. Now, lets rewrite this operator as 1st order system $$D\begin{pmatrix} u_{1}:=u \\\ u_{2}:=\partial_{x}u \\\ u_{3}:=\partial_{y}u\end{pmatrix}=\begin{pmatrix}0 &\partial_{x} &\partial_{y}\\\ \partial_{x} &-1 & 0\\\ \partial_{y} &0 &-1\end{pmatrix}\begin{pmatrix} u_{1} \\\ u_{2} \\\ u_{3}\end{pmatrix}$$

Now, the principal symbol of this operator is given by $$\sigma_{\mathrm{pr}}(x,\xi)=\begin{pmatrix}0 &\xi_{x} &\xi_{y}\\\ \xi_{x} &0 & 0\\\ \xi_{y} &0 &0\end{pmatrix}$$ which is not invertible, so $D$ is not elliptic in the usual sense. For this reason, Agmon-Douglis-Nirenberg introduced a refined definition of ellipticity, in which the fact that the entries of $D$ are of different degree is taken care of. So, one chooses integer weights $s_{i}$ for all the equations and $t_{j}$ for all the unknows such that $s_{i}+t_{j}\geq d_{ij}$ where $d_{ij}$ is the highest order of derivative of the jth unknown in the ith equation. Then, the weighted principal symbol of a differential operator $D=\sum_{\vert\alpha\vert\leq d}a_{\alpha}(x)\partial^{\alpha}$ is defined by

$${\sigma_{\mathrm{pr},w}(x,\xi)}_{ij}=\sum_{\vert\alpha\vert=s_{i}+t_{j}}(a_{\alpha}(x))_{ij}\xi^{\alpha}$$

An operator $D$ is then called "ADN-elliptic" if there exists a choice of weights such that the weighted principal symbol is invertible. Now, for the example above, suitable weights are $s_{1}=0,s_{2}=s_{3}=-1$ for equations and $t_{1}=2,t_{2}=t_{3}=1$ for unknows, so that the weighted symbol becomes $$\sigma_{\mathrm{pr},w}(x,\xi)=\begin{pmatrix}0 &\xi_{x} &\xi_{y}\\\ \xi_{x} &-1 & 0\\\ \xi_{y} &0 &-1\end{pmatrix}$$ which is invertible for $\xi\neq 0$.

What is the intuition for the choice of weights in general?

In the above example, the choice seems to be rather random. Is there some general strategy in choosing weights? In the end, the weights are there to deal with the fact that the different entries of the matrix operator $D$ are of different order. So is there a canonical choice related to the order of each entry? How does it work in the example above?