Elliptic Operator - General Definition

General Definition

Let be a (possibly nonlinear) differential operator between vector bundles of any rank. Take its principal symbol with respect to a one-form . (Basically, what we are doing is replacing the highest order covariant derivatives by vector fields .)

We say is weakly elliptic if is a linear isomorphism for every non-zero .

We say is (uniformly) strongly elliptic if for some constant ,

for all and all . It is important to note that the definition of ellipticity in the previous part of the article is strong ellipticity. Here is an inner product. Notice that the are covector fields or one-forms, but the are elements of the vector bundle upon which acts.

The quintessential example of a (strongly) elliptic operator is the Laplacian (or its negative, depending upon convention). It is not hard to see that needs to be of even order for strong ellipticity to even be an option. Otherwise, just consider plugging in both and its negative. On the other hand, a weakly elliptic first-order operator, such as the Dirac operator can square to become a strongly elliptic operator, such as the Laplacian. The composition of weakly elliptic operators is weakly elliptic.

Weak ellipticity is nevertheless strong enough for the Fredholm alternative, Schauder estimates, and the Atiyah–Singer index theorem. On the other hand, we need strong ellipticity for the maximum principle, and to guarantee that the eigenvalues are discrete, and their only limit point is infinity.