  The Hamilton-Jacobi method in holography has produced important results both
at a renormalization group (RG) fixed point and away from it. In this paper we
use the Hamilton-Jacobi method to compute the holographic trace anomaly for
four- and six-dimensional boundary conformal field theories (CFTs), assuming
higher-derivative gravity and interactions of scalar fields in the bulk. The
scalar field contributions to the anomaly appear in CFTs with exactly marginal
operators. Moving away from the fixed point, we show that the Hamilton-Jacobi
formalism provides a deep connection between the holographic and the local RG.
We derive the local RG equation holographically, and verify explicitly that it
satisfies Weyl consistency conditions stemming from the commutativity of Weyl
scalings. We also consider massive scalar fields in the bulk corresponding to
boundary relevant operators, and comment on their effects to the local RG
equation.
