  Let (M,g) be a compact Riemannian manifold with boundary. We consider the
problem (first studied by Escobar in 1992) of finding a conformal metric with
constant scalar curvature in the interior and zero mean curvature on the
boundary. Using a local test function construction, we are able to settle most
cases left open by Escobar's work. Moreover, we reduce the remaining cases to
the positive mass theorem.
