Elliptic operator

Remarks

Many authors put a factor $i^m$ in the principal symbol; this does not change the condition. For a second-order operator $L = \sum_{j,k} a_{jk}(x)\partial_j\partial_k + \sum_j b_j(x)\partial_j + c(x)$ with real symmetric $(a_{jk})$, ellipticity means that the matrix $(a_{jk}(x))$ is positive definite or negative definite at each $x$. The operator is uniformly elliptic if there is $\theta > 0$ with $\sum_{j,k} a_{jk}(x)\xi_j\xi_k \ge \theta \abs{\xi}^2$ for all $x$ and $\xi$. The definition extends to systems, where the symbol is a matrix that must be invertible, and to operators between vector bundles on manifolds.

Example: the Laplacian $\Delta = \partial_1^2 + \cdots + \partial_n^2$ has symbol $\abs{\xi}^2$, which is nonzero for $\xi \ne 0$, so it is elliptic. The Cauchy–Riemann operator $\partial_x + i\partial_y$ on $\R^2$ has symbol $\xi_1 + i\xi_2$, also nonzero for $\xi \ne 0$.

Non-example: the wave operator $\partial_1^2 - \partial_2^2$ on $\R^2$ has symbol $\xi_1^2 - \xi_2^2$, which vanishes at $\xi = (1, 1)$.