  We analyze two recently proposed methods to establish a priori lower bounds
on the minimum of general integral variational problems. The methods, which
involve either `occupation measures' or a `pointwise dual relaxation'
procedure, are shown to produce the same lower bound under a coercivity
hypothesis ensuring their strong duality. We then show by a minimax argument
that the methods actually evaluate the minimum for classes of one-dimensional,
scalar-valued, or convex multidimensional problems. For generic problems,
however, these methods should fail to capture the minimum and produce non-sharp
lower bounds. We demonstrate this using two examples, the first of which is
one-dimensional and scalar-valued with a non-convex constraint, and the second
of which is multidimensional and non-convex in a different way. The latter
example emphasizes the existence in multiple dimensions of nonlinear
constraints on gradient fields that are ignored by occupation measures, but are
built into the finer theory of gradient Young measures.
