Minimization of scalar curvature in conformal geometry

Let a smooth compact Riemannian manifold with smooth boundary and dimension . We consider a minimization problem for the scalar curvature R after a conformal change. In particular, we seek for minimizers of the functional of R, within a conformal class, under small energy assumptions and natural geometric constraints. We prove that minimizers exist, and have locally constant scalar curvature, outside of a set with explicit description.