Can 3D physics work in 2D? I mean, could 3D laws of physics exist in a 2D space? In Ads/CFT duality, for example, laws of physics in a 3D space (AdS) work in analogy with 2D laws of physics (CFT). This happens because information about the higher dimensional space is encoded in the lower dimensional one. But could this information be chamged or modified so in the lower dimensional universe there would, for example, different of exactly the same laws as the higher dimensional one (despite being in different dimensions)?
When considering different dimensions, physical laws need to be "adjusted". A typical example is any Coulomb-like force. In 3D, it scales with the inverse distance squared, but in 2D it scales with the inverse distance.
However, this is just a difference in the integral form of the equation. The local, or differential form involves the same operators, e.g.
$$\nabla\cdot\mathbf D = \rho$$
One can still argue though, that the $\nabla$ in 3D is different from $\nabla$ in 2D since these operators now act on different function spaces, so what doesn't really change is the form of these equations. Dimensionality still enters them one way or another.
I agree with Phoenix87. It is also important to note that some operations like the cross product can be generalized reasonably only to $n\geq2$, because the "$n$-dimensional cross product" maps $n-1$ vectors to one vector perpendicular to all of them.
Solving completely analogous problems (e.$\,$g. wave equations with the same source term and boundary conditions) can result in very different answers for different dimensions. There are some papers on electrodynamics in one and two dimensions which address the problems which arise reformulating the laws of physics in different spatial dimensions. They might be worth having a look at if you are interested in this.