Uniqueness of Conformal Metrics with Prescribed Scalar and mean Curvatures on Compact Manifolds with Boundary

let (mn, g) be a compact manifold with boundary and n ≥ 2. in this paper we prove the variational characterization of the neumann eigenvalues of an elliptic operator associated to the problem of conformal deformation of metrics and we study the uniqueness of metrics in the conformal class of the metric g having the same scalar curvature of the manifold and the same mean curvature of its boundary.