Working with applied math advisers as a pure math PhD student I'm currently applying to pure math PhD programs in the US to study theoretical PDE, especially things like fluids and waves. The issue I'm running into as I write applications is that at some institutions, the professors who work on this are split between the pure and applied programs. My question is whether I would be eligible to be advised by an applied professor and if I should mention applied professors in my statement. Also if this depends on the institution, how would I go about finding out in particular cases?
Here's my suggested rule of thumb:


If the pure and applied mathematicians are within the same department, mention everyone whose work you would like to contribute to - i.e. anyone who does any theoretical PDE work.
If there are separate departments, most of the people you want to work with should have a "primary" appointment in the department you are applying to. For instance, taking a look at UW's applied math department, it looks like Randy LeVeque is a professor in Applied Math, but has an adjunct appointment in pure math. This probably means his primary appointment is in Applied Math, but he could still mentor students in pure mathematics. Just be aware that if all the people you want to work for have primary appointments in different departments, it might look like you want to work in a different field (e.g. Applied Math).