  Given a compact Riemannian manifold with umbilic boundary, the Yamabe
boundary problem studies if there exist conformal scalar-flat metrics such that
the boundary has constant mean curvature. In this paper we address to the
stability of this problem with respect of perturbation of mean curvature of the
boundary and scalar curvature of the manifold. In particular we prove that the
Yamabe boundary problem is stable under perturbation of the mean curvature and
the scalar curvature from below, while it is not stable if one of the two
curvatures is perturbed from above.
