Editing Elliptic operator (section)

==General definition==

Let <math>D</math> be a (possibly nonlinear) differential operator between [[vector bundle]]s of any rank.  Take its [[Symbol of a differential operator|principal symbol]] <math>\sigma_\xi(D)</math> with respect to a one-form <math>\xi</math>.  (Basically, what we are doing is replacing the highest order [[covariant derivative]]s <math>\nabla</math> by vector fields <math>\xi</math>.)

We say <math>D</math> is ''weakly elliptic'' if <math>\sigma_\xi(D)</math> is a linear [[isomorphism]] for every non-zero <math>\xi</math>.

We say <math>D</math> is (uniformly) ''strongly elliptic'' if for some constant <math>c > 0</math>,
<math display="block">\left([\sigma_\xi(D)](v), v\right) \geq c\|v\|^2 </math>

for all <math>\|\xi\|=1</math> and all <math>v</math>.

The definition of ellipticity in the previous part of the article is ''strong ellipticity''. Here <math>(\cdot,\cdot)</math> is an inner product. Notice that the <math>\xi</math> are covector fields or one-forms, but the <math>v</math> are elements of the vector bundle upon which <math>D</math> 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 <math>D</math> needs to be of even order for strong ellipticity to even be an option.  Otherwise, just consider plugging in both <math>\xi</math> 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.