  We investigate degenerate saddle point problems, which can be viewed as limit
cases of standard mixed formulations of symmetric problems with large jumps in
coefficients. We prove that they are well-posed in a standard norm despite the
degeneracy. By wellposedness we mean a stable dependence of the solution on the
right-hand side. A known approach of splitting the saddle point problem into
separate equations for the primary unknown and for the Lagrange multiplier is
used. We revisit the traditional Ladygenskaya--Babu\v{s}ka--Brezzi (LBB) or
inf--sup condition as well as the standard coercivity condition, and analyze
how they are affected by the degeneracy of the corresponding bilinear forms. We
suggest and discuss generalized conditions that cover the degenerate case. The
LBB or inf--sup condition is necessary and sufficient for wellposedness of the
problem with respect to the Lagrange multiplier under some assumptions. The
generalized coercivity condition is necessary and sufficient for wellposedness
of the problem with respect to the primary unknown under some other
assumptions. We connect the generalized coercivity condition to the
positiveness of the minimum gap of relevant subspaces, and propose several
equivalent expressions for the minimum gap. Our results provide a foundation
for research on uniform wellposedness of mixed formulations of symmetric
problems with large jumps in coefficients in a standard norm, independent of
the jumps. Such problems appear, e.g., in numerical simulations of composite
materials made of components with contrasting properties.
