Elliptic differential operator - Imported Knowlpedia Content

Definition

Let $D:\Gamma^\infty(E)\to\Gamma^\infty(F)$ be a differential operator of order $m\geq0$ between smooth vector bundles over $M$ . The operator $D$ is elliptic if, for every $x\in M$ and every nonzero $\xi\in T_x^*M$ , its principal symbol

\sigma_m(D)(x,\xi):E_x\longrightarrow F_x

is a linear isomorphism. Ellipticity is therefore a condition on the highest-order part alone. It implies that $E$ and $F$ have equal rank on each connected component, but it does not by itself impose self-adjointness or compactness of the base manifold.

Analytic consequences

Elliptic regularity says that distributional solutions gain smoothness wherever the right-hand side is smooth. On a closed manifold, the Sobolev extension

D:H^{s+m}(E)\longrightarrow H^s(F)

is Fredholm for every real $s$ ; hence its kernel and cokernel are finite-dimensional. These consequences require analytic theorems beyond the definition and are treated in Wells, chapter IV.

Examples and non-examples

The scalar Laplace–Beltrami operator has symbol $|\xi|^2$ and is elliptic. Dirac-type operators have Clifford-linear symbol and are first-order elliptic operators. By contrast, $\partial/\partial x_1$ on $\mathbb R^n$ with $n>1$ has symbol $\xi_1$ , which vanishes for many nonzero covectors, so it is not elliptic.

Conventions and scope

References

R. O. Wells Jr., Differential Analysis on Complex Manifolds, 3rd ed., Springer, 2008. Publisher record. Relevant: chapter IV, elliptic operator theory.
H. B. Lawson Jr. and M.-L. Michelsohn, Spin Geometry, Princeton University Press, 1989. Publisher record. Relevant: chapter III, ellipticity and analytic properties.