Should I use abbreviations such as PDE or DN map in a paper? Should I write PDE or partial differential equation in the body of a mathematics paper? In a similar way, should I expand Dirichlet-to-Neumann map or write DN map or DtN map?

A somewhat related question Can I use abbreviations in the title of a research paper? considers only the title; I am more interested in the rest of the paper.
If the term is sufficiently long and sufficiently frequently used, it is reasonable to use abbrevations, as it may be very helpful for reading. I once reviewed a paper that was almost unreadable due to using some very long and complicated term all over the place (and I recommended introducing an abbrevation). For example, compare the following:

Using a solver for partial differential equations to solve a problem after applying the Dirichlet-to-Neumann map was first suggested in Ref. 42.
Using a PDE solver for a DtN-mapped problem was first suggested in Ref. 42.

However, you should explain the abbrevation the first time you are using it, e.g.:

partial differential equation (PDE)

The only exception from this are abbrevations that are extremely common like JPEG or Laser, i.e., abbrevations you could also use in the title of a paper. Depending on your field, PDE might be such an abbrevation.
As a sidenote: I recommend to decide about the usage of abbrevations and explaining them in a piecewise manner. For example if you use a term as follows:

once in section 1,
not at all in section 2,
a lot in section 3,
not at all in section 4,
a lot in section 5 and 6.

I recommend not to use or introduce the abbrevation in section 1, but only use it in sections 3, 5 and 6, introducing it when it’s first used in section 3 and 5, respectively. Depending on how likely it is that somebody reads section 6 without reading section 5, it may also be wise to introduce it again at the beginning of section 6.