Definition 12.1.5: Characteristics

Definition 12.1.5 (Characteristics).label For a linear differential operator $L=\sums{}{\abs{\al}\leq k}a_{\al}(x)\pd^{\al}$ we define the characteristic form of $L$ at $x$

\begin{align*}\chi_{L}(x,\xi)=\sums{}{\abs{\al}=k}a_{\al}(x)\xi^{\al}\end{align*}

where $\xi=(\xi_{1},...,\xi_{n})$ can be thought of as an element of the cotangent space. For example

\begin{align*}\chi_{\la}(x,\xi^{2})=\xi_{1}^{2}+\xi_{2}^{2}&&\chi_{\square}(x,\xi)=\xi_{1}^{2}-\xi_{2}^{2}\end{align*}

The characteristics of $L$ is the zero locus of the characteristic form in the projective variety

\begin{align*}char_{x}(L)=\curl{\xi\neq 0|\chi_L(x,\xi)=0}\end{align*}

We say that $L$ is elliptic at $x$ if $char_{x}(L)=\emp$. We observe that

Suppose that at some $x\in\R^{n},\xi\in char_{x}(L)$ points in $x_{j}$ directions, i.e. $\xi_{i}= 0$ for $i\neq j$ then the coefficient of $\parens{\pd_j}^{k}$ at $x$ must vanish i.e. $L$ fail to be $k$-th order in the $\xi$ direction at $x$.
One can show that $char_{x}(L)$ is defined invariantly, that is to say given a diffeomorphism $F:\OM\suf\R^{n}\to\OM'\suf\R^{n}$ with $x\mto y(x)$ then $\pd_{x}=J_{x}^{t}\pd_{y}=J^{t}_{F\inv(y)}\pd_{y}$ and $L$ gets transformed into $L'$ given by
\begin{align*}L'=\sums{}{\abs{\al}\leq k}a_{\al}(F\inv(y))(J^{t}_{F\inv(y)}\pd_{y})^{\al}\end{align*}
which has the property $char_{y}(L')$ is the image of $char_{x}(L)$. In particular, ellipticity is a coordinate-invariant property

The Eigenspace