  Given a smooth compact manifold with boundary, we study variational
properties of the volume functional and of the area functional of the boundary,
restricted to the space of the Riemannian metrics with prescribed curvature. We
obtain a sufficient and necessary condition for a metric to be a critical
point. As a by-product, a very natural analogue of V-statics metrics is
obtained. In the second part, using the Yamabe invariant in the boundary
setting, we solve the Kazdan-Warner-Kobayashi problem in a compact manifold
with boundary. For several cases, depending on the signal of the Yamabe
invariant, we give sufficient and necessary condition for a smooth function to
be the scalar or mean curvature of a metric with constraint on the volume or
area of the boundary.
