Quote:

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.

from

http://en.wikipedia.org/wiki/Elliptic_operator

Gilbarg, D.; Trudinger, Neil (1983), Elliptic Partial Differential Equations of Second Order, New York: Springer, ISBN 3-540-41160-7

Quote:

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.

from

http://en.wikipedia.org/wiki/Elliptic_operator

Gilbarg, D.; Trudinger, Neil (1983), Elliptic Partial Differential Equations of Second Order, New York: Springer, ISBN 3-540-41160-7

If you find that you are really serious about this consider contacting David R. Adams at the University of Kentucky, Lexington.