This is a classical question: how could we understand the coercivity of the operator ? Given a smooth compact Riemannian manifold of dimension , the original definition for the coercivity of such an operator is just

for any for some constant . If is a function, by taking a positive constant we immediately see that it is necessary

.

Consequently, and in particular if is constant, must be positive. If is not constant, then it is not clear to classify . In that case, we interpret this terminology in a different manner as follows

It is simple to show that the coercivity of is equivalent to . The fact that implies that the coercivity of is simple just by definition. For the reverse case, it involves a bit of calculus of variation. For interested reader, we refer to a book by O. Druet, E. Hebey, and F. Robert [here].

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