work, for simplicity we will resort to derivative expansion keeping terms upto order $k^2$. Not only will this allow us to see some of the structures emerging explicitly, but it also sets the stage for our gravitational computation in later sections.

Using the expansion above, we have $$\begin{split}
\Phi_k(r) &= \frac{\phi_k}{(2\nu)\, r^\Delta}\left(1 +\frac{2\nu}{(2\nu+2)^2} \frac{k^2}{2r^2}+\ldots \right) +   r^{\Delta-d} (J_\phi)_k \left( 1-\frac{1}{(2\nu-2)} \frac{k^2}{2r^2}+\ldots\right.\\
&\qquad\qquad \quad \left. +
\frac{\Gamma(-\nu)}{\Gamma(\nu)}\left(\frac{k}{2r}\right)^{2\nu}\left\{1 +\frac{1}{(2\nu+2)} \frac{k^2}{2r^2}+\ldots\right\}\right)
\end{split}$$ with the ellipses representing order $k^4$ terms and higher.

The source at the intermediate surface is as before easily determined $$\begin{split}
(\hat{J}_\phi)_k &\equiv \left[r^{d-\Delta}\Phi_k(r)\right]_{r\to r_D}\\
&=  (J_\phi)_k \left(1-\frac{1}{(2\nu-2)} \frac{k^2}{2\,r_D^2}+\ldots + \frac{\Gamma(-\nu)}{\Gamma(\nu)}\left(\frac{k}{2r_D}\right)^{2\nu}\left\{1 +\frac{1}{(2\nu+2)} \frac{k^2}{2\,r_D^2}+\ldots\right\}\right)\\
&\qquad \qquad +\frac{\phi_k}{(2\nu)\, r_D^{2\nu}}\left(1 +\frac{2\nu}{(2\nu+2)^2} \frac{k^2}{2\,r_D^2}+\ldots \right) 
\end{split}
\label{j2}$$ while the normalized vev of the primary to this order in derivative expansion can be determined after subtracting an appropriate counter-term as[^17] $$\begin{split}
\hat{\phi}_k &=\left[-r^{2\nu}\times r\partial_r \left( r^{d-\Delta}\Phi_k \right)+\frac{r^\Delta}{2\nu-2}\frac{k^2}{r^2}\Phi_k +\ldots \right]_{r\to r_D}\\
&=\phi_k\left(1 +\frac{2\nu-2}{(2\nu)^2} \frac{k^2}{2\,r_D^2}+\ldots \right) +\frac{4(J_\phi)_k}{(2\nu-2)} \times\frac{\Gamma(-\nu)}{\Gamma(\nu+2)}\left(\frac{k}{2}\right)^{2\nu}+\ldots  \\
\end{split}
\label{ph2}$$ To solve the Dirichlet problem,we need to solve for $\phi_k,(J_\phi)_k $ in terms of the hatted variables from and which can be inverted to get $$\begin{split}
(J_\phi)_k &=   \frac{(\hat{J}_\phi)_k}{\mathfrak{D}}\left(1 +\frac{2\nu-2}{(2\nu)^2} \frac{k^2}{2r_D^2}+\ldots \right)  -\frac{\hat{\phi}_k}{\mathfrak{D}\, (2\nu) \, r_D^{2\nu}}\left(1 +\frac{2\nu}{(2\nu+2)^2} \frac{k^2}{2\,r_D^2}+\ldots \right) \\
\phi_k &=\frac{\hat{\phi}_k}{\mathfrak{D}} \left( 1-\frac{1}{(2\nu-2)} \frac{k^2}{2\,r_D^2}+\ldots +
\frac{\Gamma(-\nu)}{\Gamma(\nu)}\left(\frac{k}{2\,r_D}\right)^{2\nu}\left\{1 +\frac{1}{(2\nu+2)} 