The key point to be aware of is that

*The holographic principle is ***not*** about plain*** restrictions*** to the boundary.*

The principle does *not* speak about relating A) fields in the bulk to B) their *restriction *to the boundary.

Instead it speaks about a more subtle and more interesting relation, namely between

A) fields in the bulk

and

B) *sources* on the boundary.

The asymptotic boundary value \(\phi|_{\partial
}\) of a field \(\phi\) in the bulk of an AdS spacetime is *not* to be identified with a field on the boundary CFT. Instead, it is to be identified with a source of the boundary CFT.

**AdS/CFT : bulk fields \(\leftrightarrow\) boundary sources**

Therefore none of the vast land of discussion of boundary value problems (which includes Stokes' and Gauss' law) is about holography.

After quantization, this relation becomes

**quantum AdS/CFT : bulk wave function \(\leftrightarrow\) boundary generating partition functions **

Just like a wave function in the bulk is a function of the fields, so a generating function for a partition function is a function of the sources: you differentiate it with respect to the sources to get the correlation functions.

Here is an example of a setup much simpler than full AdS/CFT that nevertheless does capture correctly the basic mechanism of the holographic principle (which may be what you are after):

The relation by which

1) states of 3d Chern-Simons theory

are identified with

2) pre-correlators of the 2d WZW model

("conformal blocks" namely with functions satisfying the conformal Ward identities, among which the actual acoorelators are to be found)

is an example of the holographic principle, and one that is understood at a mathematically rigorous level. Hence if you want to see a toy example in which to understand what's really going on with the holographic principle, then check out the 3dCS/2dWZW correspondence .