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