  We consider a general scalar QFT with a linear defect in $D=4-\epsilon$ and a
surface defect in $D=6-\epsilon$. Using holography and the Hamilton-Jacobi
formalism, we show that the $\beta$ functions controlling the defect RG flow
are the gradient of the entropy function. In the case of conformal field
theories, this allows the proof that the relevant $C$-functions decrease
monotonically along the RG flow. We provide evidence that this property also
holds in the full quantum theory for general scalar field theories. An
obstruction to the gradient property seems to appear at two loop order when
fermions are added.
