  In this paper, we present proofs of the coerciveness of first-order system
least-squares methods for general (possibly indefinite) second-order linear
elliptic PDEs under a minimal uniqueness assumption. For general linear
second-order elliptic PDEs, the uniqueness, existence, and well-posedness are
equivalent due to the compactness of the operator and Fredholm alternative.
Thus only a minimal uniqueness assumption is assumed: the homogeneous equation
has a unique zero solution. The coerciveness of the standard variational
problem is not required. The paper's main contribution is our first proof,
which is a straightforward and short proof using the inf-sup stability of the
standard variational formulation. The proof can potentially be applied to other
equations or settings once having the standard formulation's stability. We also
present two other proofs for the least-squares methods of general second-order
linear elliptic PDEs. The second proof is based on a lemma introduced in the
discontinuous Petrov-Galerkin method, and the third proof is based on various
stability analyses of the decomposed problems. As an application, we also
discuss least-squares finite element methods for problems with a nonsingular
$H^{-1}$ right-hand side.
